fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
right_inr (g : G) : (inr g : N ⋊[φ] G).right = g := rfl | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | right_inr | null |
inr_injective : Function.Injective (inr : G → N ⋊[φ] G) :=
Function.injective_iff_hasLeftInverse.2 ⟨right, right_inr⟩
@[simp] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | inr_injective | null |
inr_inj {g₁ g₂ : G} : (inr g₁ : N ⋊[φ] G) = inr g₂ ↔ g₁ = g₂ :=
inr_injective.eq_iff | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | inr_inj | null |
inl_aut (g : G) (n : N) : (inl (φ g n) : N ⋊[φ] G) = inr g * inl n * inr g⁻¹ := by
ext <;> simp | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | inl_aut | null |
inl_aut_inv (g : G) (n : N) : (inl ((φ g)⁻¹ n) : N ⋊[φ] G) = inr g⁻¹ * inl n * inr g := by
rw [← MonoidHom.map_inv, inl_aut, inv_inv]
@[simp] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | inl_aut_inv | null |
mk_eq_inl_mul_inr (g : G) (n : N) : (⟨n, g⟩ : N ⋊[φ] G) = inl n * inr g := by ext <;> simp
@[simp] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | mk_eq_inl_mul_inr | null |
inl_left_mul_inr_right (x : N ⋊[φ] G) : inl x.left * inr x.right = x := by ext <;> simp | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | inl_left_mul_inr_right | null |
rightHom : N ⋊[φ] G →* G where
toFun := SemidirectProduct.right
map_one' := rfl
map_mul' _ _ := rfl
@[simp] | def | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | rightHom | The canonical projection map `N ⋊[φ] G →* G`, as a group hom. |
rightHom_eq_right : (rightHom : N ⋊[φ] G → G) = right := rfl
@[simp] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | rightHom_eq_right | null |
rightHom_comp_inl : (rightHom : N ⋊[φ] G →* G).comp inl = 1 := by ext; simp [rightHom]
@[simp] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | rightHom_comp_inl | null |
rightHom_comp_inr : (rightHom : N ⋊[φ] G →* G).comp inr = MonoidHom.id _ := by
ext; simp [rightHom]
@[simp] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | rightHom_comp_inr | null |
rightHom_inl (n : N) : rightHom (inl n : N ⋊[φ] G) = 1 := by simp [rightHom]
@[simp] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | rightHom_inl | null |
rightHom_inr (g : G) : rightHom (inr g : N ⋊[φ] G) = g := by simp [rightHom] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | rightHom_inr | null |
rightHom_surjective : Function.Surjective (rightHom : N ⋊[φ] G → G) :=
Function.surjective_iff_hasRightInverse.2 ⟨inr, rightHom_inr⟩ | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | rightHom_surjective | null |
range_inl_eq_ker_rightHom : (inl : N →* N ⋊[φ] G).range = rightHom.ker :=
le_antisymm (fun _ ↦ by simp +contextual [MonoidHom.mem_ker, eq_comm])
fun x hx ↦ ⟨x.left, by ext <;> simp_all [MonoidHom.mem_ker]⟩ | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | range_inl_eq_ker_rightHom | null |
@[simps]
equivProd : N ⋊[φ] G ≃ N × G where
toFun x := ⟨x.1, x.2⟩
invFun x := ⟨x.1, x.2⟩ | def | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | equivProd | The bijection between the semidirect product and the product. |
@[simps (rhsMd := .default)]
mulEquivProd : N ⋊[1] G ≃* N × G :=
{ equivProd with map_mul' _ _ := rfl } | def | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | mulEquivProd | The group isomorphism between a semidirect product with respect to the trivial map
and the product. |
lift : N ⋊[φ] G →* H where
toFun a := fn a.1 * fg a.2
map_one' := by simp
map_mul' a b := by
have := fun n g ↦ DFunLike.ext_iff.1 (h n) g
simp only [MulAut.conj_apply, MonoidHom.comp_apply, MulEquiv.coe_toMonoidHom] at this
simp only [mul_left, mul_right, map_mul, this, mul_assoc, inv_mul_cancel_left]
@[simp] | def | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | lift | Define a group hom `N ⋊[φ] G →* H`, by defining maps `N →* H` and `G →* H` |
lift_inl (n : N) : lift fn fg h (inl n) = fn n := by simp [lift]
@[simp] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | lift_inl | null |
lift_comp_inl : (lift fn fg h).comp inl = fn := by ext; simp
@[simp] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | lift_comp_inl | null |
lift_inr (g : G) : lift fn fg h (inr g) = fg g := by simp [lift]
@[simp] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | lift_inr | null |
lift_comp_inr : (lift fn fg h).comp inr = fg := by ext; simp | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | lift_comp_inr | null |
lift_unique (F : N ⋊[φ] G →* H) :
F = lift (F.comp inl) (F.comp inr) fun _ ↦ by ext; simp [inl_aut] := by
rw [DFunLike.ext_iff]
simp only [lift, MonoidHom.comp_apply, MonoidHom.coe_mk, OneHom.coe_mk, ← map_mul,
inl_left_mul_inr_right, forall_const] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | lift_unique | null |
hom_ext {f g : N ⋊[φ] G →* H} (hl : f.comp inl = g.comp inl)
(hr : f.comp inr = g.comp inr) : f = g := by
rw [lift_unique f, lift_unique g]
simp only [*] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | hom_ext | Two maps out of the semidirect product are equal if they're equal after composition
with both `inl` and `inr` |
@[simps!]
monoidHomSubgroup {H K : Subgroup G} (h : K ≤ H.normalizer) :
H ⋊[(H.normalizerMonoidHom).comp (inclusion h)] K →* G :=
lift H.subtype K.subtype (by simp [DFunLike.ext_iff]) | def | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | monoidHomSubgroup | The homomorphism from a semidirect product of subgroups to the ambient group. |
@[simps!]
noncomputable mulEquivSubgroup {H K : Subgroup G} [H.Normal] (h : H.IsComplement' K) :
H ⋊[(H.normalizerMonoidHom).comp (inclusion (H.normalizer_eq_top ▸ le_top))] K ≃* G :=
MulEquiv.ofBijective (monoidHomSubgroup _) ((equivProd.bijective_comp _).mpr h) | def | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | mulEquivSubgroup | The isomorphism from a semidirect product of complementary subgroups to the ambient group. |
map : N₁ ⋊[φ₁] G₁ →* N₂ ⋊[φ₂] G₂ where
toFun x := ⟨fn x.1, fg x.2⟩
map_one' := by simp
map_mul' x y := by
replace h := DFunLike.ext_iff.1 (h x.right) y.left
ext <;> simp_all
@[simp] | def | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | map | Define a map from `N₁ ⋊[φ₁] G₁` to `N₂ ⋊[φ₂] G₂` given maps `N₁ →* N₂` and `G₁ →* G₂` that
satisfy a commutativity condition `∀ n g, fn (φ₁ g n) = φ₂ (fg g) (fn n)`. |
map_left (g : N₁ ⋊[φ₁] G₁) : (map fn fg h g).left = fn g.left := rfl
@[simp] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | map_left | null |
map_right (g : N₁ ⋊[φ₁] G₁) : (map fn fg h g).right = fg g.right := rfl
@[simp] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | map_right | null |
rightHom_comp_map : rightHom.comp (map fn fg h) = fg.comp rightHom := rfl
@[simp] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | rightHom_comp_map | null |
map_inl (n : N₁) : map fn fg h (inl n) = inl (fn n) := by simp [map]
@[simp] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | map_inl | null |
map_comp_inl : (map fn fg h).comp inl = inl.comp fn := by ext <;> simp
@[simp] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | map_comp_inl | null |
map_inr (g : G₁) : map fn fg h (inr g) = inr (fg g) := by simp [map]
@[simp] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | map_inr | null |
map_comp_inr : (map fn fg h).comp inr = inr.comp fg := by ext <;> simp [map] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | map_comp_inr | null |
@[simps]
congr : N₁ ⋊[φ₁] G₁ ≃* N₂ ⋊[φ₂] G₂ where
toFun x := ⟨fn x.1, fg x.2⟩
invFun x := ⟨fn.symm x.1, fg.symm x.2⟩
left_inv _ := by simp
right_inv _ := by simp
map_mul' x y := by
replace h := DFunLike.ext_iff.1 (h x.right) y.left
ext <;> simp_all | def | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | congr | Define an isomorphism from `N₁ ⋊[φ₁] G₁` to `N₂ ⋊[φ₂] G₂` given isomorphisms `N₁ ≃* N₂` and
`G₁ ≃* G₂` that satisfy a commutativity condition `∀ n g, fn (φ₁ g n) = φ₂ (fg g) (fn n)`. |
@[simps!]
congr' :
N₁ ⋊[φ₁] G₁ ≃* N₂ ⋊[MonoidHom.comp (MulAut.congr fn) (φ₁.comp fg.symm)] G₂ :=
congr fn fg (fun _ ↦ by ext; simp) | def | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | congr' | Define a isomorphism from `N₁ ⋊[φ₁] G₁` to `N₂ ⋊[φ₂] G₂` without specifying `φ₂`. |
@[simp]
card : Nat.card (N ⋊[φ] G) = Nat.card N * Nat.card G :=
Nat.card_prod _ _ ▸ Nat.card_congr equivProd | lemma | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | card | null |
derivedSeries : ℕ → Subgroup G
| 0 => ⊤
| n + 1 => ⁅derivedSeries n, derivedSeries n⁆
@[simp] | def | GroupTheory | [
"Mathlib.Data.Fin.VecNotation",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Perm.ViaEmbedding",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.SetTheory.Cardinal.Order"
] | Mathlib/GroupTheory/Solvable.lean | derivedSeries | The derived series of the group `G`, obtained by starting from the subgroup `⊤` and repeatedly
taking the commutator of the previous subgroup with itself for `n` times. |
derivedSeries_zero : derivedSeries G 0 = ⊤ :=
rfl
@[simp] | theorem | GroupTheory | [
"Mathlib.Data.Fin.VecNotation",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Perm.ViaEmbedding",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.SetTheory.Cardinal.Order"
] | Mathlib/GroupTheory/Solvable.lean | derivedSeries_zero | null |
derivedSeries_succ (n : ℕ) :
derivedSeries G (n + 1) = ⁅derivedSeries G n, derivedSeries G n⁆ :=
rfl | theorem | GroupTheory | [
"Mathlib.Data.Fin.VecNotation",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Perm.ViaEmbedding",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.SetTheory.Cardinal.Order"
] | Mathlib/GroupTheory/Solvable.lean | derivedSeries_succ | null |
derivedSeries_normal (n : ℕ) : (derivedSeries G n).Normal := by
induction n with
| zero => exact (⊤ : Subgroup G).normal_of_characteristic
| succ n ih => exact Subgroup.commutator_normal (derivedSeries G n) (derivedSeries G n)
@[simp 1100] | theorem | GroupTheory | [
"Mathlib.Data.Fin.VecNotation",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Perm.ViaEmbedding",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.SetTheory.Cardinal.Order"
] | Mathlib/GroupTheory/Solvable.lean | derivedSeries_normal | null |
derivedSeries_one : derivedSeries G 1 = commutator G :=
rfl | theorem | GroupTheory | [
"Mathlib.Data.Fin.VecNotation",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Perm.ViaEmbedding",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.SetTheory.Cardinal.Order"
] | Mathlib/GroupTheory/Solvable.lean | derivedSeries_one | null |
derivedSeries_antitone : Antitone (derivedSeries G) :=
antitone_nat_of_succ_le fun n => (derivedSeries G n).commutator_le_self | theorem | GroupTheory | [
"Mathlib.Data.Fin.VecNotation",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Perm.ViaEmbedding",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.SetTheory.Cardinal.Order"
] | Mathlib/GroupTheory/Solvable.lean | derivedSeries_antitone | null |
derivedSeries_characteristic (n : ℕ) : (derivedSeries G n).Characteristic := by
induction n with
| zero => exact Subgroup.topCharacteristic
| succ n _ => exact Subgroup.commutator_characteristic _ _ | instance | GroupTheory | [
"Mathlib.Data.Fin.VecNotation",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Perm.ViaEmbedding",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.SetTheory.Cardinal.Order"
] | Mathlib/GroupTheory/Solvable.lean | derivedSeries_characteristic | null |
map_derivedSeries_le_derivedSeries (n : ℕ) :
(derivedSeries G n).map f ≤ derivedSeries G' n := by
induction n with
| zero => exact le_top
| succ n ih => simp only [derivedSeries_succ, map_commutator, commutator_mono, ih] | theorem | GroupTheory | [
"Mathlib.Data.Fin.VecNotation",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Perm.ViaEmbedding",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.SetTheory.Cardinal.Order"
] | Mathlib/GroupTheory/Solvable.lean | map_derivedSeries_le_derivedSeries | null |
derivedSeries_le_map_derivedSeries (hf : Function.Surjective f) (n : ℕ) :
derivedSeries G' n ≤ (derivedSeries G n).map f := by
induction n with
| zero => exact (map_top_of_surjective f hf).ge
| succ n ih => exact commutator_le_map_commutator ih ih | theorem | GroupTheory | [
"Mathlib.Data.Fin.VecNotation",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Perm.ViaEmbedding",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.SetTheory.Cardinal.Order"
] | Mathlib/GroupTheory/Solvable.lean | derivedSeries_le_map_derivedSeries | null |
map_derivedSeries_eq (hf : Function.Surjective f) (n : ℕ) :
(derivedSeries G n).map f = derivedSeries G' n :=
le_antisymm (map_derivedSeries_le_derivedSeries f n) (derivedSeries_le_map_derivedSeries hf n) | theorem | GroupTheory | [
"Mathlib.Data.Fin.VecNotation",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Perm.ViaEmbedding",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.SetTheory.Cardinal.Order"
] | Mathlib/GroupTheory/Solvable.lean | map_derivedSeries_eq | null |
@[mk_iff isSolvable_def]
IsSolvable : Prop where
/-- A group `G` is solvable if its derived series is eventually trivial. -/
solvable : ∃ n : ℕ, derivedSeries G n = ⊥ | class | GroupTheory | [
"Mathlib.Data.Fin.VecNotation",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Perm.ViaEmbedding",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.SetTheory.Cardinal.Order"
] | Mathlib/GroupTheory/Solvable.lean | IsSolvable | A group `G` is solvable if its derived series is eventually trivial. We use this definition
because it's the most convenient one to work with. |
isSolvable_of_comm {G : Type*} [hG : Group G] (h : ∀ a b : G, a * b = b * a) :
IsSolvable G := by
letI hG' : CommGroup G := { hG with mul_comm := h }
cases hG
exact CommGroup.isSolvable | theorem | GroupTheory | [
"Mathlib.Data.Fin.VecNotation",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Perm.ViaEmbedding",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.SetTheory.Cardinal.Order"
] | Mathlib/GroupTheory/Solvable.lean | isSolvable_of_comm | null |
isSolvable_of_top_eq_bot (h : (⊤ : Subgroup G) = ⊥) : IsSolvable G :=
⟨⟨0, h⟩⟩ | theorem | GroupTheory | [
"Mathlib.Data.Fin.VecNotation",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Perm.ViaEmbedding",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.SetTheory.Cardinal.Order"
] | Mathlib/GroupTheory/Solvable.lean | isSolvable_of_top_eq_bot | null |
solvable_of_ker_le_range {G' G'' : Type*} [Group G'] [Group G''] (f : G' →* G)
(g : G →* G'') (hfg : g.ker ≤ f.range) [hG' : IsSolvable G'] [hG'' : IsSolvable G''] :
IsSolvable G := by
obtain ⟨n, hn⟩ := id hG''
obtain ⟨m, hm⟩ := id hG'
refine ⟨⟨n + m, le_bot_iff.mp (Subgroup.map_bot f ▸ hm ▸ ?_)⟩⟩
clear hm
induction m with
| zero =>
exact f.range_eq_map ▸ ((derivedSeries G n).map_eq_bot_iff.mp
(le_bot_iff.mp ((map_derivedSeries_le_derivedSeries g n).trans hn.le))).trans hfg
| succ m hm => exact commutator_le_map_commutator hm hm | theorem | GroupTheory | [
"Mathlib.Data.Fin.VecNotation",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Perm.ViaEmbedding",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.SetTheory.Cardinal.Order"
] | Mathlib/GroupTheory/Solvable.lean | solvable_of_ker_le_range | null |
solvable_of_solvable_injective (hf : Function.Injective f) [IsSolvable G'] :
IsSolvable G :=
solvable_of_ker_le_range (1 : G' →* G) f ((f.ker_eq_bot_iff.mpr hf).symm ▸ bot_le) | theorem | GroupTheory | [
"Mathlib.Data.Fin.VecNotation",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Perm.ViaEmbedding",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.SetTheory.Cardinal.Order"
] | Mathlib/GroupTheory/Solvable.lean | solvable_of_solvable_injective | null |
subgroup_solvable_of_solvable (H : Subgroup G) [IsSolvable G] : IsSolvable H :=
solvable_of_solvable_injective H.subtype_injective | instance | GroupTheory | [
"Mathlib.Data.Fin.VecNotation",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Perm.ViaEmbedding",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.SetTheory.Cardinal.Order"
] | Mathlib/GroupTheory/Solvable.lean | subgroup_solvable_of_solvable | null |
solvable_of_surjective (hf : Function.Surjective f) [IsSolvable G] : IsSolvable G' :=
solvable_of_ker_le_range f (1 : G' →* G) (f.range_eq_top_of_surjective hf ▸ le_top) | theorem | GroupTheory | [
"Mathlib.Data.Fin.VecNotation",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Perm.ViaEmbedding",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.SetTheory.Cardinal.Order"
] | Mathlib/GroupTheory/Solvable.lean | solvable_of_surjective | null |
solvable_quotient_of_solvable (H : Subgroup G) [H.Normal] [IsSolvable G] :
IsSolvable (G ⧸ H) :=
solvable_of_surjective (QuotientGroup.mk'_surjective H) | instance | GroupTheory | [
"Mathlib.Data.Fin.VecNotation",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Perm.ViaEmbedding",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.SetTheory.Cardinal.Order"
] | Mathlib/GroupTheory/Solvable.lean | solvable_quotient_of_solvable | null |
solvable_prod {G' : Type*} [Group G'] [IsSolvable G] [IsSolvable G'] :
IsSolvable (G × G') :=
solvable_of_ker_le_range (MonoidHom.inl G G') (MonoidHom.snd G G') fun x hx =>
⟨x.1, Prod.ext rfl hx.symm⟩
variable (G) in | instance | GroupTheory | [
"Mathlib.Data.Fin.VecNotation",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Perm.ViaEmbedding",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.SetTheory.Cardinal.Order"
] | Mathlib/GroupTheory/Solvable.lean | solvable_prod | null |
IsSolvable.commutator_lt_top_of_nontrivial [hG : IsSolvable G] [Nontrivial G] :
commutator G < ⊤ := by
rw [lt_top_iff_ne_top]
obtain ⟨n, hn⟩ := hG
contrapose! hn
refine ne_of_eq_of_ne ?_ top_ne_bot
induction n with
| zero => exact derivedSeries_zero G
| succ n h => rwa [derivedSeries_succ, h] | theorem | GroupTheory | [
"Mathlib.Data.Fin.VecNotation",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Perm.ViaEmbedding",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.SetTheory.Cardinal.Order"
] | Mathlib/GroupTheory/Solvable.lean | IsSolvable.commutator_lt_top_of_nontrivial | null |
IsSolvable.commutator_lt_of_ne_bot [IsSolvable G] {H : Subgroup G} (hH : H ≠ ⊥) :
⁅H, H⁆ < H := by
rw [← nontrivial_iff_ne_bot] at hH
rw [← H.range_subtype, MonoidHom.range_eq_map, ← map_commutator, map_subtype_lt_map_subtype]
exact commutator_lt_top_of_nontrivial H | theorem | GroupTheory | [
"Mathlib.Data.Fin.VecNotation",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Perm.ViaEmbedding",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.SetTheory.Cardinal.Order"
] | Mathlib/GroupTheory/Solvable.lean | IsSolvable.commutator_lt_of_ne_bot | null |
isSolvable_iff_commutator_lt [WellFoundedLT (Subgroup G)] :
IsSolvable G ↔ ∀ H : Subgroup G, H ≠ ⊥ → ⁅H, H⁆ < H := by
refine ⟨fun _ _ ↦ IsSolvable.commutator_lt_of_ne_bot, fun h ↦ ?_⟩
suffices h : IsSolvable (⊤ : Subgroup G) from
solvable_of_surjective (MonoidHom.range_eq_top.mp (range_subtype ⊤))
refine WellFoundedLT.induction (C := fun (H : Subgroup G) ↦ IsSolvable H) ⊤ fun H hH ↦ ?_
rcases eq_or_ne H ⊥ with rfl | h'
· infer_instance
· obtain ⟨n, hn⟩ := hH ⁅H, H⁆ (h H h')
use n + 1
rw [← (map_injective (subtype_injective _)).eq_iff, Subgroup.map_bot] at hn ⊢
rw [← hn]
clear hn
induction n with
| zero =>
rw [derivedSeries_succ, derivedSeries_zero, derivedSeries_zero, map_commutator,
← MonoidHom.range_eq_map, ← MonoidHom.range_eq_map, range_subtype, range_subtype]
| succ n ih => rw [derivedSeries_succ, map_commutator, ih, derivedSeries_succ, map_commutator] | theorem | GroupTheory | [
"Mathlib.Data.Fin.VecNotation",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Perm.ViaEmbedding",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.SetTheory.Cardinal.Order"
] | Mathlib/GroupTheory/Solvable.lean | isSolvable_iff_commutator_lt | null |
IsSimpleGroup.derivedSeries_succ {n : ℕ} : derivedSeries G n.succ = commutator G := by
induction n with
| zero => exact derivedSeries_one G
| succ n ih =>
rw [_root_.derivedSeries_succ, ih, _root_.commutator]
rcases (commutator_normal (⊤ : Subgroup G) (⊤ : Subgroup G)).eq_bot_or_eq_top with h | h
· rw [h, commutator_bot_left]
· rwa [h] | theorem | GroupTheory | [
"Mathlib.Data.Fin.VecNotation",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Perm.ViaEmbedding",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.SetTheory.Cardinal.Order"
] | Mathlib/GroupTheory/Solvable.lean | IsSimpleGroup.derivedSeries_succ | null |
IsSimpleGroup.comm_iff_isSolvable : (∀ a b : G, a * b = b * a) ↔ IsSolvable G :=
⟨isSolvable_of_comm, fun ⟨⟨n, hn⟩⟩ => by
cases n
· intro a b
refine (mem_bot.1 ?_).trans (mem_bot.1 ?_).symm <;>
· rw [← hn]
exact mem_top _
· rw [IsSimpleGroup.derivedSeries_succ] at hn
intro a b
rw [← mul_inv_eq_one, mul_inv_rev, ← mul_assoc, ← mem_bot, ← hn, commutator_eq_closure]
exact subset_closure ⟨a, b, rfl⟩⟩ | theorem | GroupTheory | [
"Mathlib.Data.Fin.VecNotation",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Perm.ViaEmbedding",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.SetTheory.Cardinal.Order"
] | Mathlib/GroupTheory/Solvable.lean | IsSimpleGroup.comm_iff_isSolvable | null |
not_solvable_of_mem_derivedSeries {g : G} (h1 : g ≠ 1)
(h2 : ∀ n : ℕ, g ∈ derivedSeries G n) : ¬IsSolvable G :=
mt (isSolvable_def _).mp
(not_exists_of_forall_not fun n h =>
h1 (Subgroup.mem_bot.mp ((congr_arg (g ∈ ·) h).mp (h2 n)))) | theorem | GroupTheory | [
"Mathlib.Data.Fin.VecNotation",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Perm.ViaEmbedding",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.SetTheory.Cardinal.Order"
] | Mathlib/GroupTheory/Solvable.lean | not_solvable_of_mem_derivedSeries | null |
Equiv.Perm.fin_5_not_solvable : ¬IsSolvable (Equiv.Perm (Fin 5)) := by
let x : Equiv.Perm (Fin 5) := ⟨![1, 2, 0, 3, 4], ![2, 0, 1, 3, 4], by decide, by decide⟩
let y : Equiv.Perm (Fin 5) := ⟨![3, 4, 2, 0, 1], ![3, 4, 2, 0, 1], by decide, by decide⟩
let z : Equiv.Perm (Fin 5) := ⟨![0, 3, 2, 1, 4], ![0, 3, 2, 1, 4], by decide, by decide⟩
have key : x = z * ⁅x, y * x * y⁻¹⁆ * z⁻¹ := by unfold x y z; decide
refine not_solvable_of_mem_derivedSeries (show x ≠ 1 by decide) fun n => ?_
induction n with
| zero => exact mem_top x
| succ n ih =>
rw [key, (derivedSeries_normal _ _).mem_comm_iff, inv_mul_cancel_left]
exact commutator_mem_commutator ih ((derivedSeries_normal _ _).conj_mem _ ih _) | theorem | GroupTheory | [
"Mathlib.Data.Fin.VecNotation",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Perm.ViaEmbedding",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.SetTheory.Cardinal.Order"
] | Mathlib/GroupTheory/Solvable.lean | Equiv.Perm.fin_5_not_solvable | null |
Equiv.Perm.not_solvable (X : Type*) (hX : 5 ≤ Cardinal.mk X) :
¬IsSolvable (Equiv.Perm X) := by
intro h
have key : Nonempty (Fin 5 ↪ X) := by
rwa [← Cardinal.lift_mk_le, Cardinal.mk_fin, Cardinal.lift_natCast, Cardinal.lift_id]
exact
Equiv.Perm.fin_5_not_solvable
(solvable_of_solvable_injective (Equiv.Perm.viaEmbeddingHom_injective (Nonempty.some key))) | theorem | GroupTheory | [
"Mathlib.Data.Fin.VecNotation",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Perm.ViaEmbedding",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.SetTheory.Cardinal.Order"
] | Mathlib/GroupTheory/Solvable.lean | Equiv.Perm.not_solvable | null |
Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
variable {p} {G} | structure | GroupTheory | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.NoncommPiCoprod"
] | Mathlib/GroupTheory/Sylow.lean | Sylow | A Sylow `p`-subgroup is a maximal `p`-subgroup. |
@[ext]
ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr | theorem | GroupTheory | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.NoncommPiCoprod"
] | Mathlib/GroupTheory/Sylow.lean | ext | null |
_root_.IsPGroup.toSylow [Fact p.Prime] {P : Subgroup G}
(hP1 : IsPGroup p P) (hP2 : ¬ p ∣ P.index) : Sylow p G :=
{ P with
isPGroup' := hP1
is_maximal' := by
intro Q hQ hPQ
have : P.FiniteIndex := ⟨fun h ↦ hP2 (h ▸ (dvd_zero p))⟩
obtain ⟨k, hk⟩ := (hQ.to_quotient (P.normalCore.subgroupOf Q)).exists_card_eq
have h := hk ▸ Nat.Prime.coprime_pow_of_not_dvd (m := k) Fact.out hP2
exact le_antisymm (Subgroup.relIndex_eq_one.mp
(Nat.eq_one_of_dvd_coprimes h (Subgroup.relIndex_dvd_index_of_le hPQ)
(Subgroup.relIndex_dvd_of_le_left Q P.normalCore_le))) hPQ }
@[simp] theorem _root_.IsPGroup.toSylow_coe [Fact p.Prime] {P : Subgroup G}
(hP1 : IsPGroup p P) (hP2 : ¬ p ∣ P.index) : (hP1.toSylow hP2) = P :=
rfl
@[simp] theorem _root_.IsPGroup.mem_toSylow [Fact p.Prime] {P : Subgroup G}
(hP1 : IsPGroup p P) (hP2 : ¬ p ∣ P.index) {g : G} : g ∈ hP1.toSylow hP2 ↔ g ∈ P :=
.rfl | def | GroupTheory | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.NoncommPiCoprod"
] | Mathlib/GroupTheory/Sylow.lean | _root_.IsPGroup.toSylow | A `p`-subgroup with index indivisible by `p` is a Sylow subgroup. |
ofCard [Finite G] {p : ℕ} [Fact p.Prime] (H : Subgroup G)
(card_eq : Nat.card H = p ^ (Nat.card G).factorization p) : Sylow p G :=
(IsPGroup.of_card card_eq).toSylow (by
rw [← mul_dvd_mul_iff_left (Nat.card_pos (α := H)).ne', card_mul_index, card_eq, ← pow_succ]
exact Nat.pow_succ_factorization_not_dvd Nat.card_pos.ne' Fact.out)
@[simp, norm_cast] | def | GroupTheory | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.NoncommPiCoprod"
] | Mathlib/GroupTheory/Sylow.lean | ofCard | A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. |
coe_ofCard [Finite G] {p : ℕ} [Fact p.Prime] (H : Subgroup G)
(card_eq : Nat.card H = p ^ (Nat.card G).factorization p) : ofCard H card_eq = H :=
rfl
variable (P : Sylow p G)
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G} | theorem | GroupTheory | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.NoncommPiCoprod"
] | Mathlib/GroupTheory/Sylow.lean | coe_ofCard | null |
comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
@[simp] | def | GroupTheory | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.NoncommPiCoprod"
] | Mathlib/GroupTheory/Sylow.lean | comapOfKerIsPGroup | The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. |
coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : P ≤ ϕ.range) :
P.comapOfKerIsPGroup ϕ hϕ h = P.comap ϕ :=
rfl | theorem | GroupTheory | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.NoncommPiCoprod"
] | Mathlib/GroupTheory/Sylow.lean | coe_comapOfKerIsPGroup | null |
comapOfInjective (hϕ : Function.Injective ϕ) (h : P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
@[simp] | def | GroupTheory | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.NoncommPiCoprod"
] | Mathlib/GroupTheory/Sylow.lean | comapOfInjective | The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. |
coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : P ≤ ϕ.range) :
P.comapOfInjective ϕ hϕ h = P.comap ϕ :=
rfl | theorem | GroupTheory | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.NoncommPiCoprod"
] | Mathlib/GroupTheory/Sylow.lean | coe_comapOfInjective | null |
protected subtype (h : P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [range_subtype])
@[simp] | def | GroupTheory | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.NoncommPiCoprod"
] | Mathlib/GroupTheory/Sylow.lean | subtype | A sylow subgroup of G is also a sylow subgroup of a subgroup of G. |
coe_subtype (h : P ≤ N) : P.subtype h = subgroupOf P N :=
rfl | theorem | GroupTheory | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.NoncommPiCoprod"
] | Mathlib/GroupTheory/Sylow.lean | coe_subtype | null |
subtype_injective {P Q : Sylow p G} {hP : P ≤ N} {hQ : Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩ | theorem | GroupTheory | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.NoncommPiCoprod"
] | Mathlib/GroupTheory/Sylow.lean | subtype_injective | null |
IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_le_nonempty₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {_} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {_} _ ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine Exists.imp (fun k hk => ?_) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM _ hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} h => ⟨⟨Q, h.2.prop, h.2.eq_of_ge⟩, h.1⟩ | theorem | GroupTheory | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.NoncommPiCoprod"
] | Mathlib/GroupTheory/Sylow.lean | IsPGroup.exists_le_sylow | A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. |
nonempty : Nonempty (Sylow p G) :=
IsPGroup.of_bot.exists_le_sylow.nonempty | instance | GroupTheory | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.NoncommPiCoprod"
] | Mathlib/GroupTheory/Sylow.lean | nonempty | null |
noncomputable inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty nonempty | instance | GroupTheory | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.NoncommPiCoprod"
] | Mathlib/GroupTheory/Sylow.lean | inhabited | null |
exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, Q.comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow | theorem | GroupTheory | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.NoncommPiCoprod"
] | Mathlib/GroupTheory/Sylow.lean | exists_comap_eq_of_ker_isPGroup | null |
exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, Q.comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf) | theorem | GroupTheory | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.NoncommPiCoprod"
] | Mathlib/GroupTheory/Sylow.lean | exists_comap_eq_of_injective | null |
exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, Q.comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective | theorem | GroupTheory | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.NoncommPiCoprod"
] | Mathlib/GroupTheory/Sylow.lean | exists_comap_subtype_eq | null |
finite_of_ker_is_pGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Finite (Sylow p G)] : Finite (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Finite.of_injective g fun P Q h => ext (by rw [← hg, h]; exact (h_exists Q).choose_spec) | theorem | GroupTheory | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.NoncommPiCoprod"
] | Mathlib/GroupTheory/Sylow.lean | finite_of_ker_is_pGroup | If the kernel of `f : H →* G` is a `p`-group,
then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. |
finite_of_injective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Finite (Sylow p G)] : Finite (Sylow p H) :=
finite_of_ker_is_pGroup (IsPGroup.ker_isPGroup_of_injective hf) | theorem | GroupTheory | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.NoncommPiCoprod"
] | Mathlib/GroupTheory/Sylow.lean | finite_of_injective | If `f : H →* G` is injective, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. |
pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨g • P.toSubgroup, P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := ext (one_smul α P.toSubgroup)
mul_smul g h P := ext (mul_smul g h P.toSubgroup) | instance | GroupTheory | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.NoncommPiCoprod"
] | Mathlib/GroupTheory/Sylow.lean | pointwiseMulAction | If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) :=
finite_of_injective H.subtype_injective
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. |
pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl | theorem | GroupTheory | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.NoncommPiCoprod"
] | Mathlib/GroupTheory/Sylow.lean | pointwise_smul_def | null |
mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj | instance | GroupTheory | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.NoncommPiCoprod"
] | Mathlib/GroupTheory/Sylow.lean | mulAction | null |
smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl | theorem | GroupTheory | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.NoncommPiCoprod"
] | Mathlib/GroupTheory/Sylow.lean | smul_def | null |
coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl | theorem | GroupTheory | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.NoncommPiCoprod"
] | Mathlib/GroupTheory/Sylow.lean | coe_subgroup_smul | null |
coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl | theorem | GroupTheory | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.NoncommPiCoprod"
] | Mathlib/GroupTheory/Sylow.lean | coe_smul | null |
smul_le {P : Sylow p G} {H : Subgroup G} (hP : P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h | theorem | GroupTheory | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.NoncommPiCoprod"
] | Mathlib/GroupTheory/Sylow.lean | smul_le | null |
smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (smul_le hP h) :=
ext (Subgroup.conj_smul_subgroupOf hP h) | theorem | GroupTheory | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.NoncommPiCoprod"
] | Mathlib/GroupTheory/Sylow.lean | smul_subtype | null |
smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ P.normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩ | theorem | GroupTheory | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.NoncommPiCoprod"
] | Mathlib/GroupTheory/Sylow.lean | smul_eq_iff_mem_normalizer | null |
smul_eq_of_normal {g : G} {P : Sylow p G} [h : P.Normal] :
g • P = P := by simp only [smul_eq_iff_mem_normalizer, P.normalizer_eq_top, mem_top] | theorem | GroupTheory | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.NoncommPiCoprod"
] | Mathlib/GroupTheory/Sylow.lean | smul_eq_of_normal | null |
Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ P.normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall | theorem | GroupTheory | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.NoncommPiCoprod"
] | Mathlib/GroupTheory/Sylow.lean | Subgroup.sylow_mem_fixedPoints_iff | null |
IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ Q.normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer) | theorem | GroupTheory | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.NoncommPiCoprod"
] | Mathlib/GroupTheory/Sylow.lean | IsPGroup.inf_normalizer_sylow | null |
IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left] | theorem | GroupTheory | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.NoncommPiCoprod"
] | Mathlib/GroupTheory/Sylow.lean | IsPGroup.sylow_mem_fixedPoints_iff | null |
Sylow.isPretransitive_of_finite [hp : Fact p.Prime] [Finite (Sylow p G)] :
IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp ?_)
calc
1 = Nat.card (fixedPoints P (orbit G P)) := ?_
_ ≡ Nat.card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Nat.card_unique (α := ({⟨P, mem_orbit_self P⟩} : Set (orbit G P))), eq_comm]
congr
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G) | instance | GroupTheory | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.NoncommPiCoprod"
] | Mathlib/GroupTheory/Sylow.lean | Sylow.isPretransitive_of_finite | A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. |
card_sylow_modEq_one [Fact p.Prime] [Finite (Sylow p G)] :
Nat.card (Sylow p G) ≡ 1 [MOD p] := by
refine Sylow.nonempty.elim fun P : Sylow p G => ?_
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have : Nat.card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this]) | theorem | GroupTheory | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.NoncommPiCoprod"
] | Mathlib/GroupTheory/Sylow.lean | card_sylow_modEq_one | A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. |
not_dvd_card_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] : ¬p ∣ Nat.card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
variable {p} {G} | theorem | GroupTheory | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.NoncommPiCoprod"
] | Mathlib/GroupTheory/Sylow.lean | not_dvd_card_sylow | null |
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