state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
case intro
p✝ : ℕ
G : Type u_1
inst✝³ : Group G
p : ℕ
inst✝² : Fact (Nat.Prime p)
N : Subgroup G
inst✝¹ : Normal N
inst✝ : Finite (Sylow p ↥N)
P : Sylow p ↥N
g : G
x✝ : g ∈ ⊤
n : ↥N
hn : n • MulAut.conjNormal g • P = P
⊢ g ∈ normalizer (map (Subgroup.subtype N) ↑P) ⊔ N | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | rw [← inv_mul_cancel_left (↑n) g, sup_comm] | /-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N =... | Mathlib.GroupTheory.Sylow.457_0.KwMUNfT2GXiDwTx | /-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N =... | Mathlib_GroupTheory_Sylow |
case intro
p✝ : ℕ
G : Type u_1
inst✝³ : Group G
p : ℕ
inst✝² : Fact (Nat.Prime p)
N : Subgroup G
inst✝¹ : Normal N
inst✝ : Finite (Sylow p ↥N)
P : Sylow p ↥N
g : G
x✝ : g ∈ ⊤
n : ↥N
hn : n • MulAut.conjNormal g • P = P
⊢ (↑n)⁻¹ * (↑n * g) ∈ N ⊔ normalizer (map (Subgroup.subtype N) ↑P) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | apply mul_mem_sup (N.inv_mem n.2) | /-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N =... | Mathlib.GroupTheory.Sylow.457_0.KwMUNfT2GXiDwTx | /-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N =... | Mathlib_GroupTheory_Sylow |
case intro
p✝ : ℕ
G : Type u_1
inst✝³ : Group G
p : ℕ
inst✝² : Fact (Nat.Prime p)
N : Subgroup G
inst✝¹ : Normal N
inst✝ : Finite (Sylow p ↥N)
P : Sylow p ↥N
g : G
x✝ : g ∈ ⊤
n : ↥N
hn : n • MulAut.conjNormal g • P = P
⊢ ↑n * g ∈ normalizer (map (Subgroup.subtype N) ↑P) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn | /-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N =... | Mathlib.GroupTheory.Sylow.457_0.KwMUNfT2GXiDwTx | /-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N =... | Mathlib_GroupTheory_Sylow |
case intro
p✝ : ℕ
G : Type u_1
inst✝³ : Group G
p : ℕ
inst✝² : Fact (Nat.Prime p)
N : Subgroup G
inst✝¹ : Normal N
inst✝ : Finite (Sylow p ↥N)
P : Sylow p ↥N
g : G
x✝ : g ∈ ⊤
n : ↥N
hn : map ((MulDistribMulAction.toMonoidEnd (MulAut ↥N) ↥N) (MulAut.conjNormal (↑n * g))) ↑P = ↑P
⊢ ↑n * g ∈ normalizer (map (Subgroup.subt... | /-
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... | refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_ | /-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N =... | Mathlib.GroupTheory.Sylow.457_0.KwMUNfT2GXiDwTx | /-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N =... | Mathlib_GroupTheory_Sylow |
case intro
p✝ : ℕ
G : Type u_1
inst✝³ : Group G
p : ℕ
inst✝² : Fact (Nat.Prime p)
N : Subgroup G
inst✝¹ : Normal N
inst✝ : Finite (Sylow p ↥N)
P : Sylow p ↥N
g : G
x✝ : g ∈ ⊤
n : ↥N
hn : map ((MulDistribMulAction.toMonoidEnd (MulAut ↥N) ↥N) (MulAut.conjNormal (↑n * g))) ↑P = ↑P
x : G
⊢ (MulEquiv.toMonoidHom (MulAut.con... | /-
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... | rw [map_map, ← congr_arg (map N.subtype) hn, map_map] | /-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N =... | Mathlib.GroupTheory.Sylow.457_0.KwMUNfT2GXiDwTx | /-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N =... | Mathlib_GroupTheory_Sylow |
case intro
p✝ : ℕ
G : Type u_1
inst✝³ : Group G
p : ℕ
inst✝² : Fact (Nat.Prime p)
N : Subgroup G
inst✝¹ : Normal N
inst✝ : Finite (Sylow p ↥N)
P : Sylow p ↥N
g : G
x✝ : g ∈ ⊤
n : ↥N
hn : map ((MulDistribMulAction.toMonoidEnd (MulAut ↥N) ↥N) (MulAut.conjNormal (↑n * g))) ↑P = ↑P
x : G
⊢ (MulEquiv.toMonoidHom (MulAut.con... | /-
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... | rfl | /-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N =... | Mathlib.GroupTheory.Sylow.457_0.KwMUNfT2GXiDwTx | /-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N =... | Mathlib_GroupTheory_Sylow |
p✝ : ℕ
G : Type u_1
inst✝³ : Group G
p : ℕ
inst✝² : Fact (Nat.Prime p)
N : Subgroup G
inst✝¹ : Normal N
inst✝ : Finite (Sylow p ↥N)
P : Sylow p G
hP : ↑P ≤ N
⊢ normalizer ↑P ⊔ N = ⊤ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP] | /-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := ... | Mathlib.GroupTheory.Sylow.477_0.KwMUNfT2GXiDwTx | /-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
s : Subgroup G
t : Set (G ⧸ s)
⊢ Fintype.card ↑(mk ⁻¹' t) = Fintype.card ↥s * Fintype.card ↑t | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)] | theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
| Mathlib.GroupTheory.Sylow.497_0.KwMUNfT2GXiDwTx | theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
H : Subgroup G
inst✝ : Finite ↑↑H
x : G
hx : ↑x ∈ MulAction.fixedPoints (↥H) (G ⧸ H)
ha : ∀ {y : G ⧸ H}, y ∈ orbit ↥H ↑x → y = ↑x
n : G
hn : n ∈ H
this : (n⁻¹ * x)⁻¹ * x ∈ H
⊢ x⁻¹ * n * x⁻¹⁻¹ ∈ H | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | rw [mul_inv_rev, inv_inv] at this | theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_... | Mathlib.GroupTheory.Sylow.503_0.KwMUNfT2GXiDwTx | theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
H : Subgroup G
inst✝ : Finite ↑↑H
x : G
hx : ↑x ∈ MulAction.fixedPoints (↥H) (G ⧸ H)
ha : ∀ {y : G ⧸ H}, y ∈ orbit ↥H ↑x → y = ↑x
n : G
hn : n ∈ H
this : x⁻¹ * n * x ∈ H
⊢ x⁻¹ * n * x⁻¹⁻¹ ∈ H | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | convert this | theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_... | Mathlib.GroupTheory.Sylow.503_0.KwMUNfT2GXiDwTx | theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H | Mathlib_GroupTheory_Sylow |
case h.e'_4.h.e'_6
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
H : Subgroup G
inst✝ : Finite ↑↑H
x : G
hx : ↑x ∈ MulAction.fixedPoints (↥H) (G ⧸ H)
ha : ∀ {y : G ⧸ H}, y ∈ orbit ↥H ↑x → y = ↑x
n : G
hn : n ∈ H
this : x⁻¹ * n * x ∈ H
⊢ x⁻¹⁻¹ = x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | rw [inv_inv] | theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_... | Mathlib.GroupTheory.Sylow.503_0.KwMUNfT2GXiDwTx | theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
H : Subgroup G
inst✝ : Finite ↑↑H
x : G
hx : ∀ (n : G), n ∈ H ↔ x * n * x⁻¹ ∈ H
y✝ : G ⧸ H
y : G
hy : Quotient.mk'' y ∈ orbit ↥H ↑x
b : G
hb₁ : b ∈ H
hb₂✝ : (fun m => m • ↑x) { val := b, property := hb₁ } = Quotient.mk'' y
hb₂ : (b * x)⁻¹ * y ∈ H
⊢ b⁻¹ * (x * (y⁻¹ * x)⁻... | /-
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... | rw [hx] at hb₂ | theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_... | Mathlib.GroupTheory.Sylow.503_0.KwMUNfT2GXiDwTx | theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
H : Subgroup G
inst✝ : Finite ↑↑H
x : G
hx : ∀ (n : G), n ∈ H ↔ x * n * x⁻¹ ∈ H
y✝ : G ⧸ H
y : G
hy : Quotient.mk'' y ∈ orbit ↥H ↑x
b : G
hb₁ : b ∈ H
hb₂✝ : (fun m => m • ↑x) { val := b, property := hb₁ } = Quotient.mk'' y
hb₂ : x * ((b * x)⁻¹ * y) * x⁻¹ ∈ H
⊢ b⁻¹ * (x ... | /-
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... | simpa [mul_inv_rev, mul_assoc] using hb₂ | theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_... | Mathlib.GroupTheory.Sylow.503_0.KwMUNfT2GXiDwTx | theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
H : Subgroup G
inst✝ : Finite ↑↑H
⊢ ∀ (x y : Subtype ↑(normalizer H)), Setoid.r x y ↔ ↑x ≈ ↑y | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | intros | /-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEqu... | Mathlib.GroupTheory.Sylow.526_0.KwMUNfT2GXiDwTx | /-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
H : Subgroup G
inst✝ : Finite ↑↑H
x✝ y✝ : Subtype ↑(normalizer H)
⊢ Setoid.r x✝ y✝ ↔ ↑x✝ ≈ ↑y✝ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | dsimp only [instHasEquiv] | /-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEqu... | Mathlib.GroupTheory.Sylow.526_0.KwMUNfT2GXiDwTx | /-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
H : Subgroup G
inst✝ : Finite ↑↑H
x✝ y✝ : Subtype ↑(normalizer H)
⊢ Setoid.r x✝ y✝ ↔ Setoid.r ↑x✝ ↑y✝ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | rw [leftRel_apply (α := normalizer H), leftRel_apply] | /-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEqu... | Mathlib.GroupTheory.Sylow.526_0.KwMUNfT2GXiDwTx | /-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
H : Subgroup G
inst✝ : Finite ↑↑H
x✝ y✝ : Subtype ↑(normalizer H)
⊢ x✝⁻¹ * y✝ ∈ comap (Subgroup.subtype (normalizer H)) H ↔ (↑x✝)⁻¹ * ↑y✝ ∈ H | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | rfl | /-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEqu... | Mathlib.GroupTheory.Sylow.526_0.KwMUNfT2GXiDwTx | /-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
⊢ Fintype.card (↥(normalizer H) ⧸ comap (Subgroup.subtype (normalizer H)) H) ≡ Fintype.card (G ⧸ H) [MOD p] | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)] | /-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subg... | Mathlib.GroupTheory.Sylow.540_0.KwMUNfT2GXiDwTx | /-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subg... | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
⊢ Fintype.card ↑(MulAction.fixedPoints (↥H) (G ⧸ H)) ≡ Fintype.card (G ⧸ H) [MOD p] | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm | /-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subg... | Mathlib.GroupTheory.Sylow.540_0.KwMUNfT2GXiDwTx | /-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subg... | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
⊢ Fintype.card ↥(normalizer H) ≡ Fintype.card G [MOD p ^ (n + 1)] | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H... | Mathlib.GroupTheory.Sylow.550_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H... | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
this : ↥(subgroupOf H (normalizer H)) ≃ ↥H
⊢ Fintype.card ↥(normalizer H) ≡ Fintype.card G [MOD p ^ (n + 1)] | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ] | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H... | Mathlib.GroupTheory.Sylow.550_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H... | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
this : ↥(subgroupOf H (normalizer H)) ≃ ↥H
⊢ Fintype.card (↥(normalizer H) ⧸ subgroupOf H (normalizer H)) * p ^ n ≡ Fintype.card (G ⧸ H) * p ^ n [MOD p * p ^ n] | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _ | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H... | Mathlib.GroupTheory.Sylow.550_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H... | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Fintype.card G
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
s : ℕ
hs : Fintype.card G = s * p ^ (n + 1)
⊢ Fintype.card (G ⧸ H) * Fintype.card ↥H = s * p * Fintype.card ↥H | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p] | /-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer ... | Mathlib.GroupTheory.Sylow.561_0.KwMUNfT2GXiDwTx | /-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer ... | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Fintype.card G
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
s : ℕ
hs : Fintype.card G = s * p ^ (n + 1)
hcard : Fintype.card (G ⧸ H) = s * p
hm : s * p % p = Fintype.card (↥(normalizer H) ⧸ comap (Sub... | /-
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... | rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm | /-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer ... | Mathlib.GroupTheory.Sylow.561_0.KwMUNfT2GXiDwTx | /-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer ... | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Fintype.card G
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
s : ℕ
hs : Fintype.card G = s * p ^ (n + 1)
⊢ Fintype.card (G ⧸ H) * Fintype.card ↥H = s * p * Fintype.card ↥H | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p] | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : F... | Mathlib.GroupTheory.Sylow.586_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : F... | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Fintype.card G
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
s : ℕ
hs : Fintype.card G = s * p ^ (n + 1)
hcard : Fintype.card (G ⧸ H) = s * p
hm : s * p % p = Fintype.card (↥(normalizer H) ⧸ subgroupOf... | /-
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... | rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : F... | Mathlib.GroupTheory.Sylow.586_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : F... | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Fintype.card G
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
s : ℕ
hs : Fintype.card G = s * p ^ (n + 1)
hcard : Fintype.card (G ⧸ H) = s * p
hm : s * p % p = Fintype.card (↥(normalizer H) ⧸ subgroupOf... | /-
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... | show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1) | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : F... | Mathlib.GroupTheory.Sylow.586_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : F... | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Fintype.card G
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
s : ℕ
hs : Fintype.card G = s * p ^ (n + 1)
hcard : Fintype.card (G ⧸ H) = s * p
hm : s * p % p = Fintype.card (↥(normalizer H) ⧸ subgroupOf... | /-
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... | suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2 | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : F... | Mathlib.GroupTheory.Sylow.586_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : F... | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Fintype.card G
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
s : ℕ
hs : Fintype.card G = s * p ^ (n + 1)
hcard : Fintype.card (G ⧸ H) = s * p
hm : s * p % p = Fintype.card (↥(normalizer H) ⧸ subgroupOf... | /-
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... | convert this using 2 | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : F... | Mathlib.GroupTheory.Sylow.586_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : F... | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Fintype.card G
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
s : ℕ
hs : Fintype.card G = s * p ^ (n + 1)
hcard : Fintype.card (G ⧸ H) = s * p
hm : s * p % p = Fintype.card (↥(normalizer H) ⧸ subgroupOf... | /-
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... | rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod] | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : F... | Mathlib.GroupTheory.Sylow.586_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : F... | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Fintype.card G
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
s : ℕ
hs : Fintype.card G = s * p ^ (n + 1)
hcard : Fintype.card (G ⧸ H) = s * p
hm : s * p % p = Fintype.card (↥(normalizer H) ⧸ subgroupOf... | /-
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... | exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)) | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : F... | Mathlib.GroupTheory.Sylow.586_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : F... | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Fintype.card G
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
s : ℕ
hs : Fintype.card G = s * p ^ (n + 1)
hcard : Fintype.card (G ⧸ H) = s * p
hm : s * p % p = Fintype.card (↥(normalizer H) ⧸ subgroupOf... | /-
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... | intro y hy | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : F... | Mathlib.GroupTheory.Sylow.586_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : F... | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Fintype.card G
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
s : ℕ
hs : Fintype.card G = s * p ^ (n + 1)
hcard : Fintype.card (G ⧸ H) = s * p
hm : s * p % p = Fintype.card (↥(normalizer H) ⧸ subgroupOf... | /-
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... | simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap] | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : F... | Mathlib.GroupTheory.Sylow.586_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : F... | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Fintype.card G
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
s : ℕ
hs : Fintype.card G = s * p ^ (n + 1)
hcard : Fintype.card (G ⧸ H) = s * p
hm : s * p % p = Fintype.card (↥(normalizer H) ⧸ subgroupOf... | /-
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... | refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩ | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : F... | Mathlib.GroupTheory.Sylow.586_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : F... | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Fintype.card G
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
s : ℕ
hs : Fintype.card G = s * p ^ (n + 1)
hcard : Fintype.card (G ⧸ H) = s * p
hm : s * p % p = Fintype.card (↥(normalizer H) ⧸ subgroupOf... | /-
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... | dsimp only | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : F... | Mathlib.GroupTheory.Sylow.586_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : F... | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Fintype.card G
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
s : ℕ
hs : Fintype.card G = s * p ^ (n + 1)
hcard : Fintype.card (G ⧸ H) = s * p
hm : s * p % p = Fintype.card (↥(normalizer H) ⧸ subgroupOf... | /-
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... | rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff] | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : F... | Mathlib.GroupTheory.Sylow.586_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : F... | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Fintype.card G
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
s : ℕ
hs : Fintype.card G = s * p ^ (n + 1)
hcard : Fintype.card (G ⧸ H) = s * p
hm : s * p % p = Fintype.card (↥(normalizer H) ⧸ subgroupOf... | /-
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... | simpa using hy | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : F... | Mathlib.GroupTheory.Sylow.586_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : F... | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p : ℕ
_hp✝ : Fact (Nat.Prime p)
n m : ℕ
hdvd : p ^ m ∣ Fintype.card G
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
hnm✝ : n ≤ m
hnm : n < m
h0m : 0 < m
_wf : m - 1 < m
hnm1 : n ≤ m - 1
K : Subgroup G
hK : Fintype.card ↥K = p ^ (m - 1) ∧ H ≤ K
⊢ p ^ (m -... | /-
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... | rwa [tsub_add_cancel_of_le h0m.nat_succ_le] | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
... | Mathlib.GroupTheory.Sylow.627_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
... | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p : ℕ
_hp✝ : Fact (Nat.Prime p)
n m : ℕ
hdvd : p ^ m ∣ Fintype.card G
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
hnm✝ : n ≤ m
hnm : n < m
h0m : 0 < m
_wf : m - 1 < m
hnm1 : n ≤ m - 1
K : Subgroup G
hK : Fintype.card ↥K = p ^ (m - 1) ∧ H ≤ K
hdvd' : p ... | /-
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... | rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le] | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
... | Mathlib.GroupTheory.Sylow.627_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
... | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p : ℕ
_hp✝ : Fact (Nat.Prime p)
n m : ℕ
hdvd : p ^ m ∣ Fintype.card G
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
hnm✝ : n ≤ m
hnm : n = m
⊢ Fintype.card ↥H = p ^ m ∧ H ≤ H | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | simp [hH, hnm] | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
... | Mathlib.GroupTheory.Sylow.627_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
... | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝² : Group G
inst✝¹ : Fintype G
p n : ℕ
inst✝ : Fact (Nat.Prime p)
hdvd : p ^ n ∣ Fintype.card G
⊢ 1 = p ^ 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | simp | /-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := ... | Mathlib.GroupTheory.Sylow.648_0.KwMUNfT2GXiDwTx | /-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝ : Group G
n p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
hn : p ^ n ≤ Nat.card G
⊢ ∃ H, Nat.card ↥H = p ^ n | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | have : Fact p.Prime := ⟨hp⟩ | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
| Mathlib.GroupTheory.Sylow.656_0.KwMUNfT2GXiDwTx | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝ : Group G
n p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
hn : p ^ n ≤ Nat.card G
this : Fact (Nat.Prime p)
⊢ ∃ H, Nat.card ↥H = p ^ n | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos] | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
hav... | Mathlib.GroupTheory.Sylow.656_0.KwMUNfT2GXiDwTx | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝ : Group G
n p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
hn : p ^ n ≤ Nat.card G
this : Fact (Nat.Prime p)
⊢ Nat.card G ≠ 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | linarith [Nat.one_le_pow n p hp.pos] | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
hav... | Mathlib.GroupTheory.Sylow.656_0.KwMUNfT2GXiDwTx | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝ : Group G
n p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
hn : p ^ n ≤ Nat.card G
this✝ : Fact (Nat.Prime p)
this : Finite G
⊢ ∃ H, Nat.card ↥H = p ^ n | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | cases nonempty_fintype G | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
hav... | Mathlib.GroupTheory.Sylow.656_0.KwMUNfT2GXiDwTx | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n | Mathlib_GroupTheory_Sylow |
case intro
G : Type u
α : Type v
β : Type w
inst✝ : Group G
n p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
hn : p ^ n ≤ Nat.card G
this✝ : Fact (Nat.Prime p)
this : Finite G
val✝ : Fintype G
⊢ ∃ H, Nat.card ↥H = p ^ n | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | obtain ⟨m, hm⟩ := h.exists_card_eq | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
hav... | Mathlib.GroupTheory.Sylow.656_0.KwMUNfT2GXiDwTx | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n | Mathlib_GroupTheory_Sylow |
case intro.intro
G : Type u
α : Type v
β : Type w
inst✝ : Group G
n p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
hn : p ^ n ≤ Nat.card G
this✝ : Fact (Nat.Prime p)
this : Finite G
val✝ : Fintype G
m : ℕ
hm : Fintype.card G = p ^ m
⊢ ∃ H, Nat.card ↥H = p ^ n | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢ | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
hav... | Mathlib.GroupTheory.Sylow.656_0.KwMUNfT2GXiDwTx | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n | Mathlib_GroupTheory_Sylow |
case intro.intro
G : Type u
α : Type v
β : Type w
inst✝ : Group G
n p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
this✝ : Fact (Nat.Prime p)
this : Finite G
val✝ : Fintype G
m : ℕ
hm : Fintype.card G = p ^ m
hn : p ^ n ≤ Fintype.card G
⊢ ∃ H, Fintype.card ↥H = p ^ n | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | refine exists_subgroup_card_pow_prime _ ?_ | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
hav... | Mathlib.GroupTheory.Sylow.656_0.KwMUNfT2GXiDwTx | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n | Mathlib_GroupTheory_Sylow |
case intro.intro
G : Type u
α : Type v
β : Type w
inst✝ : Group G
n p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
this✝ : Fact (Nat.Prime p)
this : Finite G
val✝ : Fintype G
m : ℕ
hm : Fintype.card G = p ^ m
hn : p ^ n ≤ Fintype.card G
⊢ p ^ n ∣ Fintype.card G | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | rw [hm] at hn ⊢ | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
hav... | Mathlib.GroupTheory.Sylow.656_0.KwMUNfT2GXiDwTx | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n | Mathlib_GroupTheory_Sylow |
case intro.intro
G : Type u
α : Type v
β : Type w
inst✝ : Group G
n p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
this✝ : Fact (Nat.Prime p)
this : Finite G
val✝ : Fintype G
m : ℕ
hm : Fintype.card G = p ^ m
hn : p ^ n ≤ p ^ m
⊢ p ^ n ∣ p ^ m | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
hav... | Mathlib.GroupTheory.Sylow.656_0.KwMUNfT2GXiDwTx | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝ : Group G
n p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
H : Subgroup G
hn : p ^ n ≤ Nat.card ↥H
⊢ ∃ H' ≤ H, Nat.card ↥H' = p ^ n | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ ... | Mathlib.GroupTheory.Sylow.669_0.KwMUNfT2GXiDwTx | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ ... | Mathlib_GroupTheory_Sylow |
case intro
G : Type u
α : Type v
β : Type w
inst✝ : Group G
n p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
H : Subgroup G
hn : p ^ n ≤ Nat.card ↥H
H' : Subgroup ↥H
H'card : Nat.card ↥H' = p ^ n
⊢ ∃ H' ≤ H, Nat.card ↥H' = p ^ n | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩ | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ ... | Mathlib.GroupTheory.Sylow.669_0.KwMUNfT2GXiDwTx | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ ... | Mathlib_GroupTheory_Sylow |
case intro
G : Type u
α : Type v
β : Type w
inst✝ : Group G
n p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
H : Subgroup G
hn : p ^ n ≤ Nat.card ↥H
H' : Subgroup ↥H
H'card : Nat.card ↥H' = p ^ n
⊢ Nat.card ↥(Subgroup.map (Subgroup.subtype H) H') = p ^ n | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | rw [← H'card] | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ ... | Mathlib.GroupTheory.Sylow.669_0.KwMUNfT2GXiDwTx | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ ... | Mathlib_GroupTheory_Sylow |
case intro
G : Type u
α : Type v
β : Type w
inst✝ : Group G
n p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
H : Subgroup G
hn : p ^ n ≤ Nat.card ↥H
H' : Subgroup ↥H
H'card : Nat.card ↥H' = p ^ n
⊢ Nat.card ↥(Subgroup.map (Subgroup.subtype H) H') = Nat.card ↥H' | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ ... | Mathlib.GroupTheory.Sylow.669_0.KwMUNfT2GXiDwTx | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ ... | Mathlib_GroupTheory_Sylow |
case intro
G : Type u
α : Type v
β : Type w
inst✝ : Group G
n p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
H : Subgroup G
hn : p ^ n ≤ Nat.card ↥H
H' : Subgroup ↥H
H'card : Nat.card ↥H' = p ^ n
e : ↥H' ≃* ↥(Subgroup.map (Subgroup.subtype H) H') :=
equivMapOfInjective H' (Subgroup.subtype H) (_ : Injective ⇑(Subgroup.subty... | /-
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... | exact Nat.card_congr e.symm.toEquiv | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ ... | Mathlib.GroupTheory.Sylow.669_0.KwMUNfT2GXiDwTx | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ ... | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝ : Group G
k p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
H : Subgroup G
hk : k ≤ Nat.card ↥H
hk₀ : k ≠ 0
⊢ ∃ H' ≤ H, Nat.card ↥H' ≤ k ∧ k < p * Nat.card ↥H' | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) ... | Mathlib.GroupTheory.Sylow.679_0.KwMUNfT2GXiDwTx | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) ... | Mathlib_GroupTheory_Sylow |
case intro.intro
G : Type u
α : Type v
β : Type w
inst✝ : Group G
k p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
H : Subgroup G
hk : k ≤ Nat.card ↥H
hk₀ : k ≠ 0
m : ℕ
hmk : p ^ m ≤ k
hkm : k < p ^ (m + 1)
⊢ ∃ H' ≤ H, Nat.card ↥H' ≤ k ∧ k < p * Nat.card ↥H' | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk) | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) ... | Mathlib.GroupTheory.Sylow.679_0.KwMUNfT2GXiDwTx | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) ... | Mathlib_GroupTheory_Sylow |
case intro.intro.intro.intro
G : Type u
α : Type v
β : Type w
inst✝ : Group G
k p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
H : Subgroup G
hk : k ≤ Nat.card ↥H
hk₀ : k ≠ 0
m : ℕ
hmk : p ^ m ≤ k
hkm : k < p ^ (m + 1)
H' : Subgroup G
H'H : H' ≤ H
H'card : Nat.card ↥H' = p ^ m
⊢ ∃ H' ≤ H, Nat.card ↥H' ≤ k ∧ k < p * Nat.card ↥... | /-
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... | refine ⟨H', H'H, ?_⟩ | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) ... | Mathlib.GroupTheory.Sylow.679_0.KwMUNfT2GXiDwTx | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) ... | Mathlib_GroupTheory_Sylow |
case intro.intro.intro.intro
G : Type u
α : Type v
β : Type w
inst✝ : Group G
k p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
H : Subgroup G
hk : k ≤ Nat.card ↥H
hk₀ : k ≠ 0
m : ℕ
hmk : p ^ m ≤ k
hkm : k < p ^ (m + 1)
H' : Subgroup G
H'H : H' ≤ H
H'card : Nat.card ↥H' = p ^ m
⊢ Nat.card ↥H' ≤ k ∧ k < p * Nat.card ↥H' | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | simpa only [pow_succ, H'card] using And.intro hmk hkm | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) ... | Mathlib.GroupTheory.Sylow.679_0.KwMUNfT2GXiDwTx | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) ... | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝² : Group G
inst✝¹ : Fintype G
p : ℕ
inst✝ : Fact (Nat.Prime p)
P : Sylow p G
hdvd : p ∣ Fintype.card G
⊢ p ∣ Fintype.card ↥↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | rw [← pow_one p] at hdvd | theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
| Mathlib.GroupTheory.Sylow.696_0.KwMUNfT2GXiDwTx | theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝² : Group G
inst✝¹ : Fintype G
p : ℕ
inst✝ : Fact (Nat.Prime p)
P : Sylow p G
hdvd : p ^ 1 ∣ Fintype.card G
⊢ p ∣ Fintype.card ↥↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | have key := P.pow_dvd_card_of_pow_dvd_card hdvd | theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
| Mathlib.GroupTheory.Sylow.696_0.KwMUNfT2GXiDwTx | theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝² : Group G
inst✝¹ : Fintype G
p : ℕ
inst✝ : Fact (Nat.Prime p)
P : Sylow p G
hdvd : p ^ 1 ∣ Fintype.card G
key : p ^ 1 ∣ Fintype.card ↥↑P
⊢ p ∣ Fintype.card ↥↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | rwa [pow_one] at key | theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
| Mathlib.GroupTheory.Sylow.696_0.KwMUNfT2GXiDwTx | theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p : ℕ
hp : Fact (Nat.Prime p)
P : Sylow p G
hdvd : p ∣ Fintype.card G
⊢ ↑P ≠ ⊥ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | refine' fun h => hp.out.not_dvd_one _ | theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
| Mathlib.GroupTheory.Sylow.710_0.KwMUNfT2GXiDwTx | theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p : ℕ
hp : Fact (Nat.Prime p)
P : Sylow p G
hdvd : p ∣ Fintype.card G
h : ↑P = ⊥
⊢ p ∣ 1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd | theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
| Mathlib.GroupTheory.Sylow.710_0.KwMUNfT2GXiDwTx | theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p : ℕ
hp : Fact (Nat.Prime p)
P : Sylow p G
hdvd : p ∣ Fintype.card G
h : ↑P = ⊥
key : p ∣ Fintype.card ↥↑P
⊢ p ∣ 1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | rwa [h, card_bot] at key | theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
| Mathlib.GroupTheory.Sylow.710_0.KwMUNfT2GXiDwTx | theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p : ℕ
hp : Fact (Nat.Prime p)
P : Sylow p G
⊢ Fintype.card ↥↑P = p ^ (Nat.factorization (Fintype.card G)) p | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup' | /-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
| Mathlib.GroupTheory.Sylow.717_0.KwMUNfT2GXiDwTx | /-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p | Mathlib_GroupTheory_Sylow |
case intro
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p : ℕ
hp : Fact (Nat.Prime p)
P : Sylow p G
n : ℕ
heq : Fintype.card ↥↑P = p ^ n
⊢ Fintype.card ↥↑P = p ^ (Nat.factorization (Fintype.card G)) p | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p)) | /-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
| Mathlib.GroupTheory.Sylow.717_0.KwMUNfT2GXiDwTx | /-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p | Mathlib_GroupTheory_Sylow |
case intro
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p : ℕ
hp : Fact (Nat.Prime p)
P : Sylow p G
n : ℕ
heq : Fintype.card ↥↑P = p ^ n
⊢ Fintype.card ↥↑P ∣ p ^ (Nat.factorization (Fintype.card G)) p | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | rw [heq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← heq] | /-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
... | Mathlib.GroupTheory.Sylow.717_0.KwMUNfT2GXiDwTx | /-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p | Mathlib_GroupTheory_Sylow |
case intro
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p : ℕ
hp : Fact (Nat.Prime p)
P : Sylow p G
n : ℕ
heq : Fintype.card ↥↑P = p ^ n
⊢ Fintype.card ↥↑P ∣ Fintype.card G | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | exact P.1.card_subgroup_dvd_card | /-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
... | Mathlib.GroupTheory.Sylow.717_0.KwMUNfT2GXiDwTx | /-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝³ : Group G
inst✝² : Fintype G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
H : Subgroup G
inst✝ : Fintype ↥H
card_eq : Fintype.card ↥H = p ^ (Nat.factorization (Fintype.card G)) p
⊢ ∀ {Q : Subgroup G}, IsPGroup p ↥Q → H ≤ Q → Q = H | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | obtain ⟨P, hHP⟩ := (IsPGroup.of_card card_eq).exists_le_sylow | /-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup := H
isPGroup' := IsPGroup.of_car... | Mathlib.GroupTheory.Sylow.727_0.KwMUNfT2GXiDwTx | /-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup | Mathlib_GroupTheory_Sylow |
case intro
G : Type u
α : Type v
β : Type w
inst✝³ : Group G
inst✝² : Fintype G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
H : Subgroup G
inst✝ : Fintype ↥H
card_eq : Fintype.card ↥H = p ^ (Nat.factorization (Fintype.card G)) p
P : Sylow p G
hHP : H ≤ ↑P
⊢ ∀ {Q : Subgroup G}, IsPGroup p ↥Q → H ≤ Q → Q = H | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | exact SetLike.ext'
(Set.eq_of_subset_of_card_le hHP (P.card_eq_multiplicity.trans card_eq.symm).le).symm ▸ P.3 | /-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup := H
isPGroup' := IsPGroup.of_car... | Mathlib.GroupTheory.Sylow.727_0.KwMUNfT2GXiDwTx | /-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
h : Normal ↑P
⊢ Unique (Sylow p G) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | refine { uniq := fun Q ↦ ?_ } | /-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
| Mathlib.GroupTheory.Sylow.746_0.KwMUNfT2GXiDwTx | /-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
h : Normal ↑P
Q : Sylow p G
⊢ Q = default | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | obtain ⟨x, h1⟩ := exists_smul_eq G P Q | /-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
refine { uniq := fun Q ↦ ?_ }
| Mathlib.GroupTheory.Sylow.746_0.KwMUNfT2GXiDwTx | /-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) | Mathlib_GroupTheory_Sylow |
case intro
G : Type u
α : Type v
β : Type w
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
h : Normal ↑P
Q : Sylow p G
x : G
h1 : x • P = Q
⊢ Q = default | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | obtain ⟨x, h2⟩ := exists_smul_eq G P default | /-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
refine { uniq := fun Q ↦ ?_ }
obtain ⟨x, h1⟩ := exists_smul_eq G P Q
| Mathlib.GroupTheory.Sylow.746_0.KwMUNfT2GXiDwTx | /-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) | Mathlib_GroupTheory_Sylow |
case intro.intro
G : Type u
α : Type v
β : Type w
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
h : Normal ↑P
Q : Sylow p G
x✝ : G
h1 : x✝ • P = Q
x : G
h2 : x • P = default
⊢ Q = default | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | rw [Sylow.smul_eq_of_normal] at h1 h2 | /-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
refine { uniq := fun Q ↦ ?_ }
obtain ⟨x, h1⟩ := exists_smul_eq G P Q
ob... | Mathlib.GroupTheory.Sylow.746_0.KwMUNfT2GXiDwTx | /-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) | Mathlib_GroupTheory_Sylow |
case intro.intro
G : Type u
α : Type v
β : Type w
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
h : Normal ↑P
Q : Sylow p G
x✝ : G
h1 : P = Q
x : G
h2 : P = default
⊢ Q = default | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | rw [← h1, ← h2] | /-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
refine { uniq := fun Q ↦ ?_ }
obtain ⟨x, h1⟩ := exists_smul_eq G P Q
ob... | Mathlib.GroupTheory.Sylow.746_0.KwMUNfT2GXiDwTx | /-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
h : Normal ↑P
⊢ Characteristic ↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | haveI := Sylow.unique_of_normal P h | theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
| Mathlib.GroupTheory.Sylow.760_0.KwMUNfT2GXiDwTx | theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
h : Normal ↑P
this : Unique (Sylow p G)
⊢ Characteristic ↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | rw [characteristic_iff_map_eq] | theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
haveI := Sylow.unique_of_normal P h
| Mathlib.GroupTheory.Sylow.760_0.KwMUNfT2GXiDwTx | theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
h : Normal ↑P
this : Unique (Sylow p G)
⊢ ∀ (ϕ : G ≃* G), Subgroup.map (MulEquiv.toMonoidHom ϕ) ↑P = ↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | intro Φ | theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
haveI := Sylow.unique_of_normal P h
rw [characteristic_iff_map_eq]
| Mathlib.GroupTheory.Sylow.760_0.KwMUNfT2GXiDwTx | theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
h : Normal ↑P
this : Unique (Sylow p G)
Φ : G ≃* G
⊢ Subgroup.map (MulEquiv.toMonoidHom Φ) ↑P = ↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | show (Φ • P).toSubgroup = P.toSubgroup | theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
haveI := Sylow.unique_of_normal P h
rw [characteristic_iff_map_eq]
intro Φ
| Mathlib.GroupTheory.Sylow.760_0.KwMUNfT2GXiDwTx | theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
h : Normal ↑P
this : Unique (Sylow p G)
Φ : G ≃* G
⊢ ↑(Φ • P) = ↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | congr | theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
haveI := Sylow.unique_of_normal P h
rw [characteristic_iff_map_eq]
intro Φ
show (Φ • P).toSubgroup = P.toSubgroup
| Mathlib.GroupTheory.Sylow.760_0.KwMUNfT2GXiDwTx | theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic | Mathlib_GroupTheory_Sylow |
case e_self
G : Type u
α : Type v
β : Type w
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
h : Normal ↑P
this : Unique (Sylow p G)
Φ : G ≃* G
⊢ Φ • P = P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | simp [eq_iff_true_of_subsingleton] | theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
haveI := Sylow.unique_of_normal P h
rw [characteristic_iff_map_eq]
intro Φ
show (Φ • P).toSubgroup = P.toSubgroup
congr
| Mathlib.GroupTheory.Sylow.760_0.KwMUNfT2GXiDwTx | theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
hn : Normal (normalizer ↑P)
⊢ Normal ↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | rw [← normalizer_eq_top, ← normalizer_sup_eq_top' P le_normalizer, sup_idem] | theorem normal_of_normalizer_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (↑P : Subgroup G).normalizer.Normal) : (↑P : Subgroup G).Normal := by
| Mathlib.GroupTheory.Sylow.772_0.KwMUNfT2GXiDwTx | theorem normal_of_normalizer_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (↑P : Subgroup G).normalizer.Normal) : (↑P : Subgroup G).Normal | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
⊢ normalizer (normalizer ↑P) = normalizer ↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer)) | @[simp]
theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(↑P : Subgroup G).normalizer.normalizer = (↑P : Subgroup G).normalizer := by
| Mathlib.GroupTheory.Sylow.777_0.KwMUNfT2GXiDwTx | @[simp]
theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(↑P : Subgroup G).normalizer.normalizer = (↑P : Subgroup G).normalizer | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
this :
Normal (normalizer ↑(Sylow.subtype P (_ : ↑P ≤ normalizer (normalizer ↑P)))) →
Normal ↑(Sylow.subtype P (_ : ↑P ≤ normalizer (normalizer ↑P)))
⊢ normalizer (normalizer ↑P) = normaliz... | /-
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... | simp_rw [← normalizer_eq_top, Sylow.coe_subtype, ← subgroupOf_normalizer_eq le_normalizer, ←
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this | @[simp]
theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(↑P : Subgroup G).normalizer.normalizer = (↑P : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
| Mathlib.GroupTheory.Sylow.777_0.KwMUNfT2GXiDwTx | @[simp]
theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(↑P : Subgroup G).normalizer.normalizer = (↑P : Subgroup G).normalizer | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
this : True → subgroupOf (normalizer ↑P) (normalizer (normalizer ↑P)) = ⊤
⊢ normalizer (normalizer ↑P) = normalizer ↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | rw [← subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
← this trivial] | @[simp]
theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(↑P : Subgroup G).normalizer.normalizer = (↑P : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [← normalizer_eq_top, Sylow.coe_subtype, ← ... | Mathlib.GroupTheory.Sylow.777_0.KwMUNfT2GXiDwTx | @[simp]
theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(↑P : Subgroup G).normalizer.normalizer = (↑P : Subgroup G).normalizer | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
this : True → subgroupOf (normalizer ↑P) (normalizer (normalizer ↑P)) = ⊤
⊢ Subgroup.map (Subgroup.subtype (normalizer (normalizer ↑P)))
(subgroupOf (normalizer ↑P) (normalizer (normalizer ... | /-
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... | exact map_comap_eq_self (le_normalizer.trans (ge_of_eq (subtype_range _))) | @[simp]
theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(↑P : Subgroup G).normalizer.normalizer = (↑P : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [← normalizer_eq_top, Sylow.coe_subtype, ← ... | Mathlib.GroupTheory.Sylow.777_0.KwMUNfT2GXiDwTx | @[simp]
theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(↑P : Subgroup G).normalizer.normalizer = (↑P : Subgroup G).normalizer | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝³ : Group G
inst✝² : Finite G
hnc : ∀ (H : Subgroup G), IsCoatom H → Normal H
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
⊢ normalizer ↑P = ⊤ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | rcases eq_top_or_exists_le_coatom (↑P : Subgroup G).normalizer with (heq | ⟨K, hK, hNK⟩) | theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
| Mathlib.GroupTheory.Sylow.788_0.KwMUNfT2GXiDwTx | theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal | Mathlib_GroupTheory_Sylow |
case inl
G : Type u
α : Type v
β : Type w
inst✝³ : Group G
inst✝² : Finite G
hnc : ∀ (H : Subgroup G), IsCoatom H → Normal H
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
heq : normalizer ↑P = ⊤
⊢ normalizer ↑P = ⊤ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | exact heq | theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (↑P : Subgroup G).normalizer with (heq | ⟨K, hK, ... | Mathlib.GroupTheory.Sylow.788_0.KwMUNfT2GXiDwTx | theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal | Mathlib_GroupTheory_Sylow |
case inr.intro.intro
G : Type u
α : Type v
β : Type w
inst✝³ : Group G
inst✝² : Finite G
hnc : ∀ (H : Subgroup G), IsCoatom H → Normal H
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
K : Subgroup G
hK : IsCoatom K
hNK : normalizer ↑P ≤ K
⊢ normalizer ↑P = ⊤ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | haveI := hnc _ hK | theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (↑P : Subgroup G).normalizer with (heq | ⟨K, hK, ... | Mathlib.GroupTheory.Sylow.788_0.KwMUNfT2GXiDwTx | theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal | Mathlib_GroupTheory_Sylow |
case inr.intro.intro
G : Type u
α : Type v
β : Type w
inst✝³ : Group G
inst✝² : Finite G
hnc : ∀ (H : Subgroup G), IsCoatom H → Normal H
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
K : Subgroup G
hK : IsCoatom K
hNK : normalizer ↑P ≤ K
this : Normal K
⊢ normalizer ↑P = ⊤ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | have hPK : ↑P ≤ K := le_trans le_normalizer hNK | theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (↑P : Subgroup G).normalizer with (heq | ⟨K, hK, ... | Mathlib.GroupTheory.Sylow.788_0.KwMUNfT2GXiDwTx | theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal | Mathlib_GroupTheory_Sylow |
case inr.intro.intro
G : Type u
α : Type v
β : Type w
inst✝³ : Group G
inst✝² : Finite G
hnc : ∀ (H : Subgroup G), IsCoatom H → Normal H
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
K : Subgroup G
hK : IsCoatom K
hNK : normalizer ↑P ≤ K
this : Normal K
hPK : ↑P ≤ K
⊢ normalizer ↑P = ⊤ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | refine' (hK.1 _).elim | theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (↑P : Subgroup G).normalizer with (heq | ⟨K, hK, ... | Mathlib.GroupTheory.Sylow.788_0.KwMUNfT2GXiDwTx | theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal | Mathlib_GroupTheory_Sylow |
case inr.intro.intro
G : Type u
α : Type v
β : Type w
inst✝³ : Group G
inst✝² : Finite G
hnc : ∀ (H : Subgroup G), IsCoatom H → Normal H
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
K : Subgroup G
hK : IsCoatom K
hNK : normalizer ↑P ≤ K
this : Normal K
hPK : ↑P ≤ K
⊢ K = ⊤ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | rw [← sup_of_le_right hNK, P.normalizer_sup_eq_top' hPK] | theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (↑P : Subgroup G).normalizer with (heq | ⟨K, hK, ... | Mathlib.GroupTheory.Sylow.788_0.KwMUNfT2GXiDwTx | theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
⊢ ((p : { x // x ∈ (Fintype.card G).primeFactors }) → (P : Sylow (↑p) G) → ↥↑P) ≃* G | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | set ps := (Fintype.card G).primeFactors | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
⊢ ((p : { x // x ∈ ps }) → (P : Sylow (↑p) G) → ↥↑P) ≃* G | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | let P : ∀ p, Sylow p G := default | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
⊢ ((p : { x // x ∈ ps }) → (P : Sylow (↑p) G) → ↥↑P) ≃* G | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | have hcomm : Pairwise fun p₁ p₂ : ps => ∀ x y : G, x ∈ P p₁ → y ∈ P p₂ → Commute x y := by
rintro ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ hne
haveI hp₁' := Fact.mk (Nat.prime_of_mem_primeFactors hp₁)
haveI hp₂' := Fact.mk (Nat.prime_of_mem_primeFactors hp₂)
have hne' : p₁ ≠ p₂ := by simpa using hne
apply Subgroup.commu... | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
⊢ _root_.Pairwise fun p₁ p₂ => ∀ (x y : G), x ∈ P ↑p₁ → y ∈ P ↑p₂ → Commute x y | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | rintro ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ hne | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib_GroupTheory_Sylow |
case mk.mk
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
p₁ : ℕ
hp₁ : p₁ ∈ ps
p₂ : ℕ
hp₂ : p₂ ∈ ps
hne : { val := p₁, property := hp₁ } ≠ { val :=... | /-
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... | haveI hp₁' := Fact.mk (Nat.prime_of_mem_primeFactors hp₁) | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib_GroupTheory_Sylow |
case mk.mk
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
p₁ : ℕ
hp₁ : p₁ ∈ ps
p₂ : ℕ
hp₂ : p₂ ∈ ps
hne : { val := p₁, property := hp₁ } ≠ { val :=... | /-
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... | haveI hp₂' := Fact.mk (Nat.prime_of_mem_primeFactors hp₂) | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib_GroupTheory_Sylow |
case mk.mk
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
p₁ : ℕ
hp₁ : p₁ ∈ ps
p₂ : ℕ
hp₂ : p₂ ∈ ps
hne : { val := p₁, property := hp₁ } ≠ { val :=... | /-
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... | have hne' : p₁ ≠ p₂ := by simpa using hne | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib_GroupTheory_Sylow |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.