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 ⌀ |
|---|---|---|---|---|---|---|
base_smul_def (h : H) (w : NormalWord d) :
base φ h • w = { w with head := h * w.head } := rfl | theorem | GroupTheory | [
"Mathlib.GroupTheory.CoprodI",
"Mathlib.GroupTheory.Coprod.Basic",
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/PushoutI.lean | base_smul_def | null |
summand_smul_def {i : ι} (g : G i) (w : NormalWord d) :
of (φ := φ) i g • w = (equivPair i).symm
{ equivPair i w with
head := g * (equivPair i w).head } := rfl | theorem | GroupTheory | [
"Mathlib.GroupTheory.CoprodI",
"Mathlib.GroupTheory.Coprod.Basic",
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/PushoutI.lean | summand_smul_def | null |
of_smul_eq_smul {i : ι} (g : G i) (w : NormalWord d) :
of (φ := φ) i g • w = g • w := by
rw [summand_smul_def, summand_smul_def'] | theorem | GroupTheory | [
"Mathlib.GroupTheory.CoprodI",
"Mathlib.GroupTheory.Coprod.Basic",
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/PushoutI.lean | of_smul_eq_smul | null |
base_smul_eq_smul (h : H) (w : NormalWord d) :
base φ h • w = h • w := by
rw [base_smul_def, base_smul_def'] | theorem | GroupTheory | [
"Mathlib.GroupTheory.CoprodI",
"Mathlib.GroupTheory.Coprod.Basic",
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/PushoutI.lean | base_smul_eq_smul | null |
@[elab_as_elim]
noncomputable consRecOn {motive : NormalWord d → Sort _} (w : NormalWord d)
(empty : motive empty)
(cons : ∀ (i : ι) (g : G i) (w : NormalWord d) (hmw : w.fstIdx ≠ some i)
(_hgn : g ∈ d.set i) (hgr : g ∉ (φ i).range) (_hw1 : w.head = 1),
motive w → motive (cons g w hmw hgr))
(bas... | def | GroupTheory | [
"Mathlib.GroupTheory.CoprodI",
"Mathlib.GroupTheory.Coprod.Basic",
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/PushoutI.lean | consRecOn | Induction principle for `NormalWord`, that corresponds closely to inducting on
the underlying list. |
cons_eq_smul {i : ι} (g : G i)
(w : NormalWord d) (hmw : w.fstIdx ≠ some i)
(hgr : g ∉ (φ i).range) : cons g w hmw hgr = of (φ := φ) i g • w := by
apply ext_smul i
simp only [cons, Word.cons_eq_smul, MonoidHom.apply_ofInjective_symm,
equiv_fst_eq_mul_inv, mul_assoc, map_mul, map_inv, mul_smul, inv_smul... | theorem | GroupTheory | [
"Mathlib.GroupTheory.CoprodI",
"Mathlib.GroupTheory.Coprod.Basic",
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/PushoutI.lean | cons_eq_smul | null |
prod_summand_smul {i : ι} (g : G i) (w : NormalWord d) :
(g • w).prod = of i g * w.prod := by
simp only [prod, summand_smul_def', equivPair, rcons, Word.equivPair_symm,
Equiv.coe_fn_mk, Equiv.coe_fn_symm_mk, Word.equivPair_smul_same,
Word.equivPair_tail_eq_inv_smul, Word.rcons_eq_smul, ← of_apply_eq_base ... | theorem | GroupTheory | [
"Mathlib.GroupTheory.CoprodI",
"Mathlib.GroupTheory.Coprod.Basic",
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/PushoutI.lean | prod_summand_smul | null |
prod_smul (g : PushoutI φ) (w : NormalWord d) :
(g • w).prod = g * w.prod := by
induction g using PushoutI.induction_on generalizing w with
| of i g => rw [of_smul_eq_smul, prod_summand_smul]
| base h => rw [base_smul_eq_smul, prod_base_smul]
| mul x y ihx ihy => rw [mul_smul, ihx, ihy, mul_assoc] | theorem | GroupTheory | [
"Mathlib.GroupTheory.CoprodI",
"Mathlib.GroupTheory.Coprod.Basic",
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/PushoutI.lean | prod_smul | null |
prod_smul_empty (w : NormalWord d) : w.prod • empty = w := by
induction w using consRecOn with
| empty => simp
| cons i g w _ _ _ _ ih =>
rw [prod_cons, mul_smul, ih, cons_eq_smul]
| base h w _ ih =>
rw [prod_smul, mul_smul, ih] | theorem | GroupTheory | [
"Mathlib.GroupTheory.CoprodI",
"Mathlib.GroupTheory.Coprod.Basic",
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/PushoutI.lean | prod_smul_empty | null |
noncomputable equiv : PushoutI φ ≃ NormalWord d :=
{ toFun := fun g => g • .empty
invFun := fun w => w.prod
left_inv := fun g => by
simp only [prod_smul, prod_empty, mul_one]
right_inv := fun w => prod_smul_empty w } | def | GroupTheory | [
"Mathlib.GroupTheory.CoprodI",
"Mathlib.GroupTheory.Coprod.Basic",
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/PushoutI.lean | equiv | The equivalence between normal forms and elements of the pushout |
prod_injective {ι : Type*} {G : ι → Type*} [(i : ι) → Group (G i)] {φ : (i : ι) → H →* G i}
{d : Transversal φ} : Function.Injective (prod : NormalWord d → PushoutI φ) := by
letI := Classical.decEq ι
letI := fun i => Classical.decEq (G i)
classical exact equiv.symm.injective | theorem | GroupTheory | [
"Mathlib.GroupTheory.CoprodI",
"Mathlib.GroupTheory.Coprod.Basic",
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/PushoutI.lean | prod_injective | null |
of_injective (hφ : ∀ i, Function.Injective (φ i)) (i : ι) :
Function.Injective (of (φ := φ) i) := by
rcases transversal_nonempty φ hφ with ⟨d⟩
let _ := Classical.decEq ι
let _ := fun i => Classical.decEq (G i)
refine Function.Injective.of_comp
(f := ((· • ·) : PushoutI φ → NormalWord d → NormalWord d)) ... | theorem | GroupTheory | [
"Mathlib.GroupTheory.CoprodI",
"Mathlib.GroupTheory.Coprod.Basic",
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/PushoutI.lean | of_injective | All maps into the `PushoutI`, or amalgamated product of groups are injective,
provided all maps in the diagram are injective.
See also `base_injective` |
base_injective (hφ : ∀ i, Function.Injective (φ i)) :
Function.Injective (base φ) := by
rcases transversal_nonempty φ hφ with ⟨d⟩
let _ := Classical.decEq ι
let _ := fun i => Classical.decEq (G i)
refine Function.Injective.of_comp
(f := ((· • ·) : PushoutI φ → NormalWord d → NormalWord d)) ?_
intro _ ... | theorem | GroupTheory | [
"Mathlib.GroupTheory.CoprodI",
"Mathlib.GroupTheory.Coprod.Basic",
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/PushoutI.lean | base_injective | null |
Reduced (w : Word G) : Prop :=
∀ g, g ∈ w.toList → g.2 ∉ (φ g.1).range | def | GroupTheory | [
"Mathlib.GroupTheory.CoprodI",
"Mathlib.GroupTheory.Coprod.Basic",
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/PushoutI.lean | Reduced | A word in `CoprodI` is reduced if none of its letters are in the base group. |
Reduced.exists_normalWord_prod_eq (d : Transversal φ) {w : Word G} (hw : Reduced φ w) :
∃ w' : NormalWord d, w'.prod = ofCoprodI w.prod ∧
w'.toList.map Sigma.fst = w.toList.map Sigma.fst := by
classical
induction w using Word.consRecOn with
| empty => exact ⟨empty, by simp, rfl⟩
| cons i g w hIdx hg1 ... | theorem | GroupTheory | [
"Mathlib.GroupTheory.CoprodI",
"Mathlib.GroupTheory.Coprod.Basic",
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/PushoutI.lean | Reduced.exists_normalWord_prod_eq | null |
Reduced.eq_empty_of_mem_range
(hφ : ∀ i, Injective (φ i)) {w : Word G} (hw : Reduced φ w)
(h : ofCoprodI w.prod ∈ (base φ).range) : w = .empty := by
rcases transversal_nonempty φ hφ with ⟨d⟩
rcases hw.exists_normalWord_prod_eq d with ⟨w', hw'prod, hw'map⟩
rcases h with ⟨h, heq⟩
have : (NormalWord.prod (... | theorem | GroupTheory | [
"Mathlib.GroupTheory.CoprodI",
"Mathlib.GroupTheory.Coprod.Basic",
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/PushoutI.lean | Reduced.eq_empty_of_mem_range | For any word `w` in the coproduct,
if `w` is reduced (i.e none its letters are in the image of the base monoid), and nonempty, then
`w` itself is not in the image of the base group. |
inf_of_range_eq_base_range
(hφ : ∀ i, Injective (φ i)) {i j : ι} (hij : i ≠ j) :
(of i).range ⊓ (of j).range = (base φ).range :=
le_antisymm
(by
intro x ⟨⟨g₁, hg₁⟩, ⟨g₂, hg₂⟩⟩
by_contra hx
have hx1 : x ≠ 1 := by rintro rfl; simp_all only [ne_eq, one_mem, not_true_eq_false]
have hg₁... | theorem | GroupTheory | [
"Mathlib.GroupTheory.CoprodI",
"Mathlib.GroupTheory.Coprod.Basic",
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/PushoutI.lean | inf_of_range_eq_base_range | The intersection of the images of the maps from any two distinct groups in the diagram
into the amalgamated product is the image of the map from the base group in the diagram. |
@[to_additive /-- The minimum number of generators of an additive group. -/]
noncomputable rank [h : FG G] : ℕ := @Nat.find _ (Classical.decPred _) (fg_iff'.mp h)
variable (G) in
@[to_additive] | def | GroupTheory | [
"Mathlib.GroupTheory.Finiteness",
"Mathlib.SetTheory.Cardinal.Finite"
] | Mathlib/GroupTheory/Rank.lean | rank | The minimum number of generators of a group. |
rank_spec [h : FG G] : ∃ S : Finset G, S.card = rank G ∧ .closure S = (⊤ : Subgroup G) :=
@Nat.find_spec _ (Classical.decPred _) (fg_iff'.mp h)
@[to_additive] | lemma | GroupTheory | [
"Mathlib.GroupTheory.Finiteness",
"Mathlib.SetTheory.Cardinal.Finite"
] | Mathlib/GroupTheory/Rank.lean | rank_spec | null |
rank_le [h : FG G] {S : Finset G} (hS : .closure S = (⊤ : Subgroup G)) : rank G ≤ S.card :=
@Nat.find_le _ _ (Classical.decPred _) (fg_iff'.mp h) ⟨S, rfl, hS⟩
@[to_additive] | lemma | GroupTheory | [
"Mathlib.GroupTheory.Finiteness",
"Mathlib.SetTheory.Cardinal.Finite"
] | Mathlib/GroupTheory/Rank.lean | rank_le | null |
rank_le_of_surjective [FG G] [FG H] (f : G →* H) (hf : Surjective f) : rank H ≤ rank G := by
classical
obtain ⟨S, hS1, hS2⟩ := rank_spec G
trans (S.image f).card
· apply rank_le
rw [Finset.coe_image, ← MonoidHom.map_closure, hS2, Subgroup.map_top_of_surjective f hf]
· exact Finset.card_image_le.trans_eq h... | lemma | GroupTheory | [
"Mathlib.GroupTheory.Finiteness",
"Mathlib.SetTheory.Cardinal.Finite"
] | Mathlib/GroupTheory/Rank.lean | rank_le_of_surjective | null |
rank_range_le [FG G] {f : G →* H} : rank f.range ≤ rank G :=
rank_le_of_surjective f.rangeRestrict f.rangeRestrict_surjective
@[to_additive] | lemma | GroupTheory | [
"Mathlib.GroupTheory.Finiteness",
"Mathlib.SetTheory.Cardinal.Finite"
] | Mathlib/GroupTheory/Rank.lean | rank_range_le | null |
rank_congr [FG G] [FG H] (e : G ≃* H) : rank G = rank H :=
le_antisymm (rank_le_of_surjective e.symm e.symm.surjective)
(rank_le_of_surjective e e.surjective) | lemma | GroupTheory | [
"Mathlib.GroupTheory.Finiteness",
"Mathlib.SetTheory.Cardinal.Finite"
] | Mathlib/GroupTheory/Rank.lean | rank_congr | null |
@[to_additive]
rank_congr {H K : Subgroup G} [Group.FG H] [Group.FG K] (h : H = K) : rank H = rank K := by
subst h; rfl
@[to_additive] | lemma | GroupTheory | [
"Mathlib.GroupTheory.Finiteness",
"Mathlib.SetTheory.Cardinal.Finite"
] | Mathlib/GroupTheory/Rank.lean | rank_congr | null |
rank_closure_finset_le_card (s : Finset G) : rank (closure (s : Set G)) ≤ s.card := by
classical
let t : Finset (closure (s : Set G)) := s.preimage Subtype.val Subtype.coe_injective.injOn
have ht : closure (t : Set (closure (s : Set G))) = ⊤ := by
rw [Finset.coe_preimage]
exact closure_preimage_eq_top (s ... | lemma | GroupTheory | [
"Mathlib.GroupTheory.Finiteness",
"Mathlib.SetTheory.Cardinal.Finite"
] | Mathlib/GroupTheory/Rank.lean | rank_closure_finset_le_card | null |
rank_closure_finite_le_nat_card (s : Set G) [Finite s] : rank (closure s) ≤ Nat.card s := by
haveI := Fintype.ofFinite s
rw [Nat.card_eq_fintype_card, ← s.toFinset_card, ← rank_congr (congr_arg _ s.coe_toFinset)]
exact rank_closure_finset_le_card s.toFinset | lemma | GroupTheory | [
"Mathlib.GroupTheory.Finiteness",
"Mathlib.SetTheory.Cardinal.Finite"
] | Mathlib/GroupTheory/Rank.lean | rank_closure_finite_le_nat_card | null |
nat_card_centralizer_nat_card_stabilizer (g : G) :
Nat.card (centralizer {g}) = Nat.card (MulAction.stabilizer (ConjAct G) g) := by
rw [centralizer_eq_comap_stabilizer]; rfl | lemma | GroupTheory | [
"Mathlib.GroupTheory.Finiteness",
"Mathlib.SetTheory.Cardinal.Finite"
] | Mathlib/GroupTheory/Rank.lean | nat_card_centralizer_nat_card_stabilizer | null |
@[ext]
RegularWreathProduct where
/-- The function of Q → D -/
left : Q → D
/-- The element of Q -/
right : Q
@[inherit_doc] infix:65 " ≀ᵣ " => RegularWreathProduct | structure | GroupTheory | [
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.PUnit",
"Mathlib.Data.Finite.Perm",
"Mathlib.Algebra.Group.End"
] | Mathlib/GroupTheory/RegularWreathProduct.lean | RegularWreathProduct | The regular wreath product of groups `Q` and `D`. It is the product `(Q → D) × Q` with the group
operation `⟨a₁, a₂⟩ * ⟨b₁, b₂⟩ = ⟨a₁ * (fun x ↦ b₁ (a₂⁻¹ * x)), a₂ * b₂⟩`. |
mul_def (a b : D ≀ᵣ Q) : a * b = ⟨a.1 * fun x ↦ b.1 (a.2⁻¹ * x), a.2 * b.2⟩ := rfl
@[simp] | lemma | GroupTheory | [
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.PUnit",
"Mathlib.Data.Finite.Perm",
"Mathlib.Algebra.Group.End"
] | Mathlib/GroupTheory/RegularWreathProduct.lean | mul_def | null |
mul_left (a b : D ≀ᵣ Q) : (a * b).1 = a.1 * fun x ↦ b.1 (a.2⁻¹ * x) := rfl
@[simp] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.PUnit",
"Mathlib.Data.Finite.Perm",
"Mathlib.Algebra.Group.End"
] | Mathlib/GroupTheory/RegularWreathProduct.lean | mul_left | null |
mul_right (a b : D ≀ᵣ Q) : (a * b).right = a.right * b.right := rfl | theorem | GroupTheory | [
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.PUnit",
"Mathlib.Data.Finite.Perm",
"Mathlib.Algebra.Group.End"
] | Mathlib/GroupTheory/RegularWreathProduct.lean | mul_right | null |
@[simp]
one_left : (1 : D ≀ᵣ Q).left = 1 := rfl
@[simp] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.PUnit",
"Mathlib.Data.Finite.Perm",
"Mathlib.Algebra.Group.End"
] | Mathlib/GroupTheory/RegularWreathProduct.lean | one_left | null |
one_right : (1 : D ≀ᵣ Q).right = 1 := rfl | theorem | GroupTheory | [
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.PUnit",
"Mathlib.Data.Finite.Perm",
"Mathlib.Algebra.Group.End"
] | Mathlib/GroupTheory/RegularWreathProduct.lean | one_right | null |
@[simp]
inv_left (a : D ≀ᵣ Q) : a⁻¹.left = fun x ↦ a.left⁻¹ (a.right * x) := rfl
@[simp] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.PUnit",
"Mathlib.Data.Finite.Perm",
"Mathlib.Algebra.Group.End"
] | Mathlib/GroupTheory/RegularWreathProduct.lean | inv_left | null |
inv_right (a : D ≀ᵣ Q) : a⁻¹.right = a.right⁻¹ := rfl | theorem | GroupTheory | [
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.PUnit",
"Mathlib.Data.Finite.Perm",
"Mathlib.Algebra.Group.End"
] | Mathlib/GroupTheory/RegularWreathProduct.lean | inv_right | null |
rightHom : D ≀ᵣ Q →* Q where
toFun := RegularWreathProduct.right
map_one' := rfl
map_mul' _ _ := rfl | def | GroupTheory | [
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.PUnit",
"Mathlib.Data.Finite.Perm",
"Mathlib.Algebra.Group.End"
] | Mathlib/GroupTheory/RegularWreathProduct.lean | rightHom | The canonical projection map `D ≀ᵣ Q →* Q`, as a group hom. |
inl : Q →* D ≀ᵣ Q where
toFun q := ⟨1, q⟩
map_one' := rfl
map_mul' _ _ := by ext <;> simp
@[simp] | def | GroupTheory | [
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.PUnit",
"Mathlib.Data.Finite.Perm",
"Mathlib.Algebra.Group.End"
] | Mathlib/GroupTheory/RegularWreathProduct.lean | inl | The canonical map `Q →* D ≀ᵣ Q` sending `q` to `⟨1, q⟩` |
left_inl (q : Q) : (inl q : D ≀ᵣ Q).left = 1 := rfl
@[simp] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.PUnit",
"Mathlib.Data.Finite.Perm",
"Mathlib.Algebra.Group.End"
] | Mathlib/GroupTheory/RegularWreathProduct.lean | left_inl | null |
right_inl (q : Q) : (inl q : D ≀ᵣ Q).right = q := rfl
@[simp] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.PUnit",
"Mathlib.Data.Finite.Perm",
"Mathlib.Algebra.Group.End"
] | Mathlib/GroupTheory/RegularWreathProduct.lean | right_inl | null |
rightHom_eq_right : (rightHom : D ≀ᵣ Q → Q) = right := rfl
@[simp] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.PUnit",
"Mathlib.Data.Finite.Perm",
"Mathlib.Algebra.Group.End"
] | Mathlib/GroupTheory/RegularWreathProduct.lean | rightHom_eq_right | null |
rightHom_comp_inl_eq_id : (rightHom : D ≀ᵣ Q →* Q).comp inl = MonoidHom.id _ := by ext; simp
@[simp] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.PUnit",
"Mathlib.Data.Finite.Perm",
"Mathlib.Algebra.Group.End"
] | Mathlib/GroupTheory/RegularWreathProduct.lean | rightHom_comp_inl_eq_id | null |
fun_id (q : Q) : rightHom (inl q : D ≀ᵣ Q) = q := by simp | theorem | GroupTheory | [
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.PUnit",
"Mathlib.Data.Finite.Perm",
"Mathlib.Algebra.Group.End"
] | Mathlib/GroupTheory/RegularWreathProduct.lean | fun_id | null |
equivProd D Q : D ≀ᵣ Q ≃ (Q → D) × Q where
toFun := fun ⟨d, q⟩ => ⟨d, q⟩
invFun := fun ⟨d, q⟩ => ⟨d, q⟩
left_inv := fun _ => rfl
right_inv := fun _ => rfl | def | GroupTheory | [
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.PUnit",
"Mathlib.Data.Finite.Perm",
"Mathlib.Algebra.Group.End"
] | Mathlib/GroupTheory/RegularWreathProduct.lean | equivProd | The equivalence map for the representation as a product. |
card [Finite Q] : Nat.card (D ≀ᵣ Q) = Nat.card D ^ Nat.card Q * Nat.card Q := by
rw [Nat.card_congr (equivProd D Q), Nat.card_prod (Q → D) Q, Nat.card_fun] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.PUnit",
"Mathlib.Data.Finite.Perm",
"Mathlib.Algebra.Group.End"
] | Mathlib/GroupTheory/RegularWreathProduct.lean | card | null |
congr {D₁ Q₁ D₂ Q₂ : Type*} [Group D₁] [Group Q₁] [Group D₂] [Group Q₂]
(f : D₁ ≃* D₂) (g : Q₁ ≃* Q₂) :
D₁ ≀ᵣ Q₁ ≃* D₂ ≀ᵣ Q₂ where
toFun x := ⟨f ∘ (x.left ∘ g.symm), g x.right⟩
invFun x := ⟨(f.symm ∘ x.left) ∘ g, g.symm x.right⟩
left_inv x := by ext <;> simp
right_inv x := by ext <;> simp
map_mul' x y... | def | GroupTheory | [
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.PUnit",
"Mathlib.Data.Finite.Perm",
"Mathlib.Algebra.Group.End"
] | Mathlib/GroupTheory/RegularWreathProduct.lean | congr | Define an isomorphism from `D₁ ≀ᵣ Q₁` to `D₂ ≀ᵣ Q₂`
given isomorphisms `D₁ ≀ᵣ Q₁` and `Q₁ ≃* Q₂`. |
@[simp]
smul_def {w : D ≀ᵣ Q} {p : Λ × Q} : w • p = ⟨(w.1 (w.2 * p.2)) • p.1, w.2 * p.2⟩ := rfl | lemma | GroupTheory | [
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.PUnit",
"Mathlib.Data.Finite.Perm",
"Mathlib.Algebra.Group.End"
] | Mathlib/GroupTheory/RegularWreathProduct.lean | smul_def | null |
toPerm : D ≀ᵣ Q →* Equiv.Perm (Λ × Q) :=
MulAction.toPermHom (D ≀ᵣ Q) (Λ × Q) | def | GroupTheory | [
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.PUnit",
"Mathlib.Data.Finite.Perm",
"Mathlib.Algebra.Group.End"
] | Mathlib/GroupTheory/RegularWreathProduct.lean | toPerm | The map sending the wreath product `D ≀ᵣ Q` to its representation as a permutation of `Λ × Q`
given `D`-set `Λ`. |
toPermInj [Nonempty Λ] : Function.Injective (toPerm D Q Λ) := MulAction.toPerm_injective | theorem | GroupTheory | [
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.PUnit",
"Mathlib.Data.Finite.Perm",
"Mathlib.Algebra.Group.End"
] | Mathlib/GroupTheory/RegularWreathProduct.lean | toPermInj | null |
IteratedWreathProduct (G : Type u) : (n : ℕ) → Type u
| 0 => PUnit
| n + 1 => (IteratedWreathProduct G n) ≀ᵣ G
variable (G : Type u) (n : ℕ)
@[simp] | def | GroupTheory | [
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.PUnit",
"Mathlib.Data.Finite.Perm",
"Mathlib.Algebra.Group.End"
] | Mathlib/GroupTheory/RegularWreathProduct.lean | IteratedWreathProduct | The wreath product of group `G` iterated `n` times. |
IteratedWreathProduct_zero : IteratedWreathProduct G 0 = PUnit := rfl
@[simp] | lemma | GroupTheory | [
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.PUnit",
"Mathlib.Data.Finite.Perm",
"Mathlib.Algebra.Group.End"
] | Mathlib/GroupTheory/RegularWreathProduct.lean | IteratedWreathProduct_zero | null |
IteratedWreathProduct_succ :
IteratedWreathProduct G (n + 1) = (IteratedWreathProduct G n) ≀ᵣ G := rfl | lemma | GroupTheory | [
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.PUnit",
"Mathlib.Data.Finite.Perm",
"Mathlib.Algebra.Group.End"
] | Mathlib/GroupTheory/RegularWreathProduct.lean | IteratedWreathProduct_succ | null |
IteratedWreathProduct.card [Finite G] : Nat.card (IteratedWreathProduct G n) =
Nat.card G ^ (∑ i ∈ Finset.range n, Nat.card G ^ i) := by
induction n with
| zero => simp
| succ n h => rw [IteratedWreathProduct_succ, RegularWreathProduct.card,
h, geom_sum_succ, pow_succ, pow_mul']
variable [Group G] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.PUnit",
"Mathlib.Data.Finite.Perm",
"Mathlib.Algebra.Group.End"
] | Mathlib/GroupTheory/RegularWreathProduct.lean | IteratedWreathProduct.card | null |
iteratedWreathToPermHom (G : Type*) [Group G] :
(n : ℕ) → (IteratedWreathProduct G n →* Equiv.Perm (Fin n → G))
| 0 => 1
| n + 1 => by
let _ := MulAction.compHom (Fin n → G) (iteratedWreathToPermHom G n)
exact (Fin.succFunEquiv G n).symm.permCongrHom.toMonoidHom.comp
(RegularWreathProduct.to... | def | GroupTheory | [
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.PUnit",
"Mathlib.Data.Finite.Perm",
"Mathlib.Algebra.Group.End"
] | Mathlib/GroupTheory/RegularWreathProduct.lean | iteratedWreathToPermHom | The homomorphism from `IteratedWreathProduct G n` to `Perm (Fin n → G)`. |
iteratedWreathToPermHomInj (G : Type*) [Group G] :
(n : ℕ) → Function.Injective (iteratedWreathToPermHom G n)
| 0 => by
simp only [IteratedWreathProduct_zero]
apply Function.injective_of_subsingleton
| n + 1 => by
let _ := MulAction.compHom (Fin n → G) (iteratedWreathToPermHom G n)
have ... | lemma | GroupTheory | [
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.PUnit",
"Mathlib.Data.Finite.Perm",
"Mathlib.Algebra.Group.End"
] | Mathlib/GroupTheory/RegularWreathProduct.lean | iteratedWreathToPermHomInj | null |
noncomputable Sylow.mulEquivIteratedWreathProduct (p : ℕ) [hp : Fact (Nat.Prime p)] (n : ℕ)
(α : Type*) [Finite α] (hα : Nat.card α = p ^ n)
(G : Type*) [Group G] [Finite G] (hG : Nat.card G = p)
(P : Sylow p (Equiv.Perm α)) :
P ≃* IteratedWreathProduct G n := by
let e1 : α ≃ (Fin n → G) := (Finite.eq... | def | GroupTheory | [
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.PUnit",
"Mathlib.Data.Finite.Perm",
"Mathlib.Algebra.Group.End"
] | Mathlib/GroupTheory/RegularWreathProduct.lean | Sylow.mulEquivIteratedWreathProduct | The encoding of the Sylow `p`-subgroups of `Perm α` as an iterated wreath product. |
@[to_additive]
card_dvd_exponent_pow_rank : Nat.card G ∣ Monoid.exponent G ^ Group.rank G := by
classical
obtain ⟨S, hS1, hS2⟩ := Group.rank_spec G
rw [← hS1, ← Fintype.card_coe, ← Finset.card_univ, ← Finset.prod_const]
let f : (∀ g : S, zpowers (g : G)) →* G := noncommPiCoprod fun s t _ x y _ _ => mul_comm x _... | theorem | GroupTheory | [
"Mathlib.Algebra.Group.Pointwise.Finset.Basic",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Commutator.Finite",
"Mathlib.GroupTheory.Transfer"
] | Mathlib/GroupTheory/Schreier.lean | card_dvd_exponent_pow_rank | null |
card_dvd_exponent_pow_rank' {n : ℕ} (hG : ∀ g : G, g ^ n = 1) :
Nat.card G ∣ n ^ Group.rank G :=
(card_dvd_exponent_pow_rank G).trans
(pow_dvd_pow_of_dvd (Monoid.exponent_dvd_of_forall_pow_eq_one hG) (Group.rank G)) | theorem | GroupTheory | [
"Mathlib.Algebra.Group.Pointwise.Finset.Basic",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Commutator.Finite",
"Mathlib.GroupTheory.Transfer"
] | Mathlib/GroupTheory/Schreier.lean | card_dvd_exponent_pow_rank' | null |
closure_mul_image_mul_eq_top
(hR : IsComplement H R) (hR1 : (1 : G) ∈ R) (hS : closure S = ⊤) :
(closure ((R * S).image fun g => g * (hR.toRightFun g : G)⁻¹)) * R = ⊤ := by
let f : G → R := hR.toRightFun
let U : Set G := (R * S).image fun g => g * (f g : G)⁻¹
change (closure U : Set G) * R = ⊤
refine to... | theorem | GroupTheory | [
"Mathlib.Algebra.Group.Pointwise.Finset.Basic",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Commutator.Finite",
"Mathlib.GroupTheory.Transfer"
] | Mathlib/GroupTheory/Schreier.lean | closure_mul_image_mul_eq_top | null |
closure_mul_image_eq (hR : IsComplement H R) (hR1 : (1 : G) ∈ R)
(hS : closure S = ⊤) : closure ((R * S).image fun g => g * (hR.toRightFun g : G)⁻¹) = H := by
have hU : closure ((R * S).image fun g => g * (hR.toRightFun g : G)⁻¹) ≤ H := by
rw [closure_le]
rintro - ⟨g, -, rfl⟩
exact hR.mul_inv_toRightF... | theorem | GroupTheory | [
"Mathlib.Algebra.Group.Pointwise.Finset.Basic",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Commutator.Finite",
"Mathlib.GroupTheory.Transfer"
] | Mathlib/GroupTheory/Schreier.lean | closure_mul_image_eq | **Schreier's Lemma**: If `R : Set G` is a `rightTransversal` of `H : Subgroup G`
with `1 ∈ R`, and if `G` is generated by `S : Set G`, then `H` is generated by the `Set`
`(R * S).image (fun g ↦ g * (hR.toRightFun g)⁻¹)`. |
closure_mul_image_eq_top (hR : IsComplement H R) (hR1 : (1 : G) ∈ R)
(hS : closure S = ⊤) : closure ((R * S).image fun g =>
⟨g * (hR.toRightFun g : G)⁻¹, hR.mul_inv_toRightFun_mem g⟩ : Set H) = ⊤ := by
rw [eq_top_iff, ← map_subtype_le_map_subtype, MonoidHom.map_closure, Set.image_image]
exact (map_subtype... | theorem | GroupTheory | [
"Mathlib.Algebra.Group.Pointwise.Finset.Basic",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Commutator.Finite",
"Mathlib.GroupTheory.Transfer"
] | Mathlib/GroupTheory/Schreier.lean | closure_mul_image_eq_top | **Schreier's Lemma**: If `R : Set G` is a `rightTransversal` of `H : Subgroup G`
with `1 ∈ R`, and if `G` is generated by `S : Set G`, then `H` is generated by the `Set`
`(R * S).image (fun g ↦ g * (hR.toRightFun g)⁻¹)`. |
closure_mul_image_eq_top' [DecidableEq G] {R S : Finset G}
(hR : IsComplement (H : Set G) R) (hR1 : (1 : G) ∈ R)
(hS : closure (S : Set G) = ⊤) :
closure (((R * S).image fun g => ⟨_, hR.mul_inv_toRightFun_mem g⟩ : Finset H) : Set H) = ⊤ := by
rw [Finset.coe_image, Finset.coe_mul]
exact closure_mul_image... | theorem | GroupTheory | [
"Mathlib.Algebra.Group.Pointwise.Finset.Basic",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Commutator.Finite",
"Mathlib.GroupTheory.Transfer"
] | Mathlib/GroupTheory/Schreier.lean | closure_mul_image_eq_top' | **Schreier's Lemma**: If `R : Finset G` is a `rightTransversal` of `H : Subgroup G`
with `1 ∈ R`, and if `G` is generated by `S : Finset G`, then `H` is generated by the `Finset`
`(R * S).image (fun g ↦ g * (hR.toRightFun g)⁻¹)`. |
exists_finset_card_le_mul [FiniteIndex H] {S : Finset G} (hS : closure (S : Set G) = ⊤) :
∃ T : Finset H, #T ≤ H.index * #S ∧ closure (T : Set H) = ⊤ := by
letI := H.fintypeQuotientOfFiniteIndex
haveI : DecidableEq G := Classical.decEq G
obtain ⟨R₀, hR, hR1⟩ := H.exists_isComplement_right 1
haveI : Fintype ... | theorem | GroupTheory | [
"Mathlib.Algebra.Group.Pointwise.Finset.Basic",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Commutator.Finite",
"Mathlib.GroupTheory.Transfer"
] | Mathlib/GroupTheory/Schreier.lean | exists_finset_card_le_mul | null |
fg_of_index_ne_zero [hG : Group.FG G] [FiniteIndex H] : Group.FG H := by
obtain ⟨S, hS⟩ := hG.1
obtain ⟨T, -, hT⟩ := exists_finset_card_le_mul H hS
exact ⟨⟨T, hT⟩⟩ | instance | GroupTheory | [
"Mathlib.Algebra.Group.Pointwise.Finset.Basic",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Commutator.Finite",
"Mathlib.GroupTheory.Transfer"
] | Mathlib/GroupTheory/Schreier.lean | fg_of_index_ne_zero | **Schreier's Lemma**: A finite index subgroup of a finitely generated
group is finitely generated. |
rank_le_index_mul_rank [hG : Group.FG G] [FiniteIndex H] :
Group.rank H ≤ H.index * Group.rank G := by
haveI := H.fg_of_index_ne_zero
obtain ⟨S, hS₀, hS⟩ := Group.rank_spec G
obtain ⟨T, hT₀, hT⟩ := exists_finset_card_le_mul H hS
calc
Group.rank H ≤ #T := Group.rank_le hT
_ ≤ H.index * #S := hT₀
... | theorem | GroupTheory | [
"Mathlib.Algebra.Group.Pointwise.Finset.Basic",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Commutator.Finite",
"Mathlib.GroupTheory.Transfer"
] | Mathlib/GroupTheory/Schreier.lean | rank_le_index_mul_rank | null |
card_commutator_dvd_index_center_pow [Finite (commutatorSet G)] :
Nat.card (_root_.commutator G) ∣
(center G).index ^ ((center G).index * Nat.card (commutatorSet G) + 1) := by
by_cases hG : (center G).index = 0
· simp_rw [hG, zero_mul, zero_add, pow_one, dvd_zero]
haveI : FiniteIndex (center G) := ⟨hG⟩
... | theorem | GroupTheory | [
"Mathlib.Algebra.Group.Pointwise.Finset.Basic",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Commutator.Finite",
"Mathlib.GroupTheory.Transfer"
] | Mathlib/GroupTheory/Schreier.lean | card_commutator_dvd_index_center_pow | If `G` has `n` commutators `[g₁, g₂]`, then `|G'| ∣ [G : Z(G)] ^ ([G : Z(G)] * n + 1)`,
where `G'` denotes the commutator of `G`. |
cardCommutatorBound (n : ℕ) :=
(n ^ (2 * n)) ^ (n ^ (2 * n + 1) + 1) | def | GroupTheory | [
"Mathlib.Algebra.Group.Pointwise.Finset.Basic",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Commutator.Finite",
"Mathlib.GroupTheory.Transfer"
] | Mathlib/GroupTheory/Schreier.lean | cardCommutatorBound | A bound for the size of the commutator subgroup in terms of the number of commutators. |
card_commutator_le_of_finite_commutatorSet [Finite (commutatorSet G)] :
Nat.card (_root_.commutator G) ≤ cardCommutatorBound (Nat.card (commutatorSet G)) := by
have h1 := index_center_le_pow (closureCommutatorRepresentatives G)
have h2 := card_commutator_dvd_index_center_pow (closureCommutatorRepresentatives G)... | theorem | GroupTheory | [
"Mathlib.Algebra.Group.Pointwise.Finset.Basic",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.GroupTheory.Commutator.Finite",
"Mathlib.GroupTheory.Transfer"
] | Mathlib/GroupTheory/Schreier.lean | card_commutator_le_of_finite_commutatorSet | A theorem of Schur: The size of the commutator subgroup is bounded in terms of the number of
commutators. |
QuotientDiff :=
Quotient
(Setoid.mk (fun α β => diff (MonoidHom.id H) α β = 1)
⟨fun α => diff_self (MonoidHom.id H) α, fun h => by rw [← diff_inv, h, inv_one],
fun h h' => by rw [← diff_mul_diff, h, h', one_mul]⟩) | def | GroupTheory | [
"Mathlib.GroupTheory.Transfer"
] | Mathlib/GroupTheory/SchurZassenhaus.lean | QuotientDiff | The quotient of the transversals of an abelian normal `N` by the `diff` relation. |
smul_diff_smul' [hH : Normal H] (g : Gᵐᵒᵖ) :
diff (MonoidHom.id H) (g • α) (g • β) =
⟨g.unop⁻¹ * (diff (MonoidHom.id H) α β : H) * g.unop,
hH.mem_comm ((congr_arg (· ∈ H) (mul_inv_cancel_left _ _)).mpr (SetLike.coe_mem _))⟩ := by
letI := H.fintypeQuotientOfFiniteIndex
let ϕ : H →* H :=
{ toFun... | theorem | GroupTheory | [
"Mathlib.GroupTheory.Transfer"
] | Mathlib/GroupTheory/SchurZassenhaus.lean | smul_diff_smul' | null |
smul_diff' (h : H) :
diff (MonoidHom.id H) α (op (h : G) • β) = diff (MonoidHom.id H) α β * h ^ H.index := by
letI := H.fintypeQuotientOfFiniteIndex
rw [diff, diff, index_eq_card, Nat.card_eq_fintype_card,
← Finset.card_univ, ← Finset.prod_const, ← Finset.prod_mul_distrib]
refine Finset.prod_congr rfl f... | theorem | GroupTheory | [
"Mathlib.GroupTheory.Transfer"
] | Mathlib/GroupTheory/SchurZassenhaus.lean | smul_diff' | null |
eq_one_of_smul_eq_one (hH : Nat.Coprime (Nat.card H) H.index) (α : H.QuotientDiff)
(h : H) : h • α = α → h = 1 :=
Quotient.inductionOn' α fun α hα =>
(powCoprime hH).injective <|
calc
h ^ H.index = diff (MonoidHom.id H) (op ((h⁻¹ : H) : G) • α) α := by
rw [← diff_inv, smul_diff', diff_... | theorem | GroupTheory | [
"Mathlib.GroupTheory.Transfer"
] | Mathlib/GroupTheory/SchurZassenhaus.lean | eq_one_of_smul_eq_one | null |
exists_smul_eq (hH : Nat.Coprime (Nat.card H) H.index) (α β : H.QuotientDiff) :
∃ h : H, h • α = β :=
Quotient.inductionOn' α
(Quotient.inductionOn' β fun β α =>
Exists.imp (fun _ => Quotient.sound')
⟨(powCoprime hH).symm (diff (MonoidHom.id H) β α),
(diff_inv _ _ _).symm.trans
... | theorem | GroupTheory | [
"Mathlib.GroupTheory.Transfer"
] | Mathlib/GroupTheory/SchurZassenhaus.lean | exists_smul_eq | null |
isComplement'_stabilizer_of_coprime {α : H.QuotientDiff}
(hH : Nat.Coprime (Nat.card H) H.index) : IsComplement' H (stabilizer G α) :=
isComplement'_stabilizer α (eq_one_of_smul_eq_one hH α) fun g => exists_smul_eq hH (g • α) α | theorem | GroupTheory | [
"Mathlib.GroupTheory.Transfer"
] | Mathlib/GroupTheory/SchurZassenhaus.lean | isComplement'_stabilizer_of_coprime | null |
private exists_right_complement'_of_coprime_aux (hH : Nat.Coprime (Nat.card H) H.index) :
∃ K : Subgroup G, IsComplement' H K :=
have ne : Nonempty (QuotientDiff H) := inferInstance
ne.elim fun α => ⟨stabilizer G α, isComplement'_stabilizer_of_coprime hH⟩ | theorem | GroupTheory | [
"Mathlib.GroupTheory.Transfer"
] | Mathlib/GroupTheory/SchurZassenhaus.lean | exists_right_complement'_of_coprime_aux | Do not use this lemma: It is made obsolete by `exists_right_complement'_of_coprime` |
private step0 : N ≠ ⊥ := by
rintro rfl
exact h3 ⊤ isComplement'_bot_top
variable [Finite G]
include h2 in | theorem | GroupTheory | [
"Mathlib.GroupTheory.Transfer"
] | Mathlib/GroupTheory/SchurZassenhaus.lean | step0 | Do not use this lemma: It is made obsolete by `exists_right_complement'_of_coprime` |
private step1 (K : Subgroup G) (hK : K ⊔ N = ⊤) : K = ⊤ := by
contrapose! h3
have h4 : (N.comap K.subtype).index = N.index := by
rw [← N.relIndex_top_right, ← hK]
exact (relIndex_sup_right K N).symm
have h5 : Nat.card K < Nat.card G := by
rw [← K.index_mul_card]
exact lt_mul_of_one_lt_left Nat.car... | theorem | GroupTheory | [
"Mathlib.GroupTheory.Transfer"
] | Mathlib/GroupTheory/SchurZassenhaus.lean | step1 | Do not use this lemma: It is made obsolete by `exists_right_complement'_of_coprime` |
private step2 (K : Subgroup G) [K.Normal] (hK : K ≤ N) : K = ⊥ ∨ K = N := by
have : Function.Surjective (QuotientGroup.mk' K) := Quotient.mk''_surjective
have h4 := step1 h1 h2 h3
contrapose! h4
have h5 : Nat.card (G ⧸ K) < Nat.card G := by
rw [← index_eq_card, ← K.index_mul_card]
refine
lt_mul_of... | theorem | GroupTheory | [
"Mathlib.GroupTheory.Transfer"
] | Mathlib/GroupTheory/SchurZassenhaus.lean | step2 | Do not use this lemma: It is made obsolete by `exists_right_complement'_of_coprime` |
private step3 (K : Subgroup N) [(K.map N.subtype).Normal] : K = ⊥ ∨ K = ⊤ := by
have key := step2 h1 h2 h3 (K.map N.subtype) (map_subtype_le K)
rw [← map_bot N.subtype] at key
conv at key =>
rhs
rhs
rw [← N.range_subtype, N.subtype.range_eq_map]
have inj := map_injective N.subtype_injective
rwa [i... | theorem | GroupTheory | [
"Mathlib.GroupTheory.Transfer"
] | Mathlib/GroupTheory/SchurZassenhaus.lean | step3 | Do not use this lemma: It is made obsolete by `exists_right_complement'_of_coprime` |
private step4 : (Nat.card N).minFac.Prime :=
Nat.minFac_prime (N.one_lt_card_iff_ne_bot.mpr (step0 h1 h3)).ne' | theorem | GroupTheory | [
"Mathlib.GroupTheory.Transfer"
] | Mathlib/GroupTheory/SchurZassenhaus.lean | step4 | Do not use this lemma: It is made obsolete by `exists_right_complement'_of_coprime` |
private step5 {P : Sylow (Nat.card N).minFac N} : P.1 ≠ ⊥ := by
haveI : Fact (Nat.card N).minFac.Prime := ⟨step4 h1 h3⟩
apply P.ne_bot_of_dvd_card
exact (Nat.card N).minFac_dvd
include h2 in | theorem | GroupTheory | [
"Mathlib.GroupTheory.Transfer"
] | Mathlib/GroupTheory/SchurZassenhaus.lean | step5 | Do not use this lemma: It is made obsolete by `exists_right_complement'_of_coprime` |
private step6 : IsPGroup (Nat.card N).minFac N := by
haveI : Fact (Nat.card N).minFac.Prime := ⟨step4 h1 h3⟩
refine Sylow.nonempty.elim fun P => P.2.of_surjective P.1.subtype ?_
rw [← MonoidHom.range_eq_top, range_subtype]
haveI : (P.1.map N.subtype).Normal :=
normalizer_eq_top_iff.mp (step1 h1 h2 h3 (P.map... | theorem | GroupTheory | [
"Mathlib.GroupTheory.Transfer"
] | Mathlib/GroupTheory/SchurZassenhaus.lean | step6 | Do not use this lemma: It is made obsolete by `exists_right_complement'_of_coprime` |
step7 : IsMulCommutative N := by
haveI := N.bot_or_nontrivial.resolve_left (step0 h1 h3)
haveI : Fact (Nat.card N).minFac.Prime := ⟨step4 h1 h3⟩
exact
⟨⟨fun g h => ((eq_top_iff.mp ((step3 h1 h2 h3 (center N)).resolve_left
(step6 h1 h2 h3).bot_lt_center.ne') (mem_top h)).comm g).symm⟩⟩ | theorem | GroupTheory | [
"Mathlib.GroupTheory.Transfer"
] | Mathlib/GroupTheory/SchurZassenhaus.lean | step7 | Do not use this lemma: It is made obsolete by `exists_right_complement'_of_coprime` |
private exists_right_complement'_of_coprime_aux' [Finite G] (hG : Nat.card G = n)
{N : Subgroup G} [N.Normal] (hN : Nat.Coprime (Nat.card N) N.index) :
∃ H : Subgroup G, IsComplement' N H := by
revert G
induction n using Nat.strongRecOn with | ind n ih => ?_
rintro G _ _ rfl N _ hN
refine not_forall_not... | theorem | GroupTheory | [
"Mathlib.GroupTheory.Transfer"
] | Mathlib/GroupTheory/SchurZassenhaus.lean | exists_right_complement'_of_coprime_aux' | Do not use this lemma: It is made obsolete by `exists_right_complement'_of_coprime` |
exists_right_complement'_of_coprime {N : Subgroup G} [N.Normal]
(hN : Nat.Coprime (Nat.card N) N.index) : ∃ H : Subgroup G, IsComplement' N H := by
by_cases hN1 : Nat.card N = 0
· rw [hN1, Nat.coprime_zero_left, index_eq_one] at hN
rw [hN]
exact ⟨⊥, isComplement'_top_bot⟩
by_cases hN2 : N.index = 0
... | theorem | GroupTheory | [
"Mathlib.GroupTheory.Transfer"
] | Mathlib/GroupTheory/SchurZassenhaus.lean | exists_right_complement'_of_coprime | **Schur-Zassenhaus** for normal subgroups:
If `H : Subgroup G` is normal, and has order coprime to its index, then there exists a
subgroup `K` which is a (right) complement of `H`. |
exists_left_complement'_of_coprime {N : Subgroup G} [N.Normal]
(hN : Nat.Coprime (Nat.card N) N.index) : ∃ H : Subgroup G, IsComplement' H N :=
Exists.imp (fun _ => IsComplement'.symm) (exists_right_complement'_of_coprime hN) | theorem | GroupTheory | [
"Mathlib.GroupTheory.Transfer"
] | Mathlib/GroupTheory/SchurZassenhaus.lean | exists_left_complement'_of_coprime | **Schur-Zassenhaus** for normal subgroups:
If `H : Subgroup G` is normal, and has order coprime to its index, then there exists a
subgroup `K` which is a (left) complement of `H`. |
@[ext]
SemidirectProduct (φ : G →* MulAut N) where
/-- The element of N -/
left : N
/-- The element of G -/
right : G
deriving DecidableEq
attribute [pp_using_anonymous_constructor] SemidirectProduct
@[inherit_doc]
notation:35 N " ⋊[" φ:35 "] " G:35 => SemidirectProduct N G φ | structure | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | SemidirectProduct | The semidirect product of groups `N` and `G`, given a map `φ` from `G` to the automorphism
group of `N`. It the product of sets with the group operation
`⟨n₁, g₁⟩ * ⟨n₂, g₂⟩ = ⟨n₁ * φ g₁ n₂, g₁ * g₂⟩` |
mul_def (a b : SemidirectProduct N G φ) : a * b = ⟨a.1 * φ a.2 b.1, a.2 * b.2⟩ := rfl
@[simp] | lemma | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | mul_def | null |
mul_left (a b : N ⋊[φ] G) : (a * b).left = a.left * φ a.right b.left := rfl
@[simp] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | mul_left | null |
mul_right (a b : N ⋊[φ] G) : (a * b).right = a.right * b.right := rfl | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | mul_right | null |
@[simp]
one_left : (1 : N ⋊[φ] G).left = 1 := rfl
@[simp] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | one_left | null |
one_right : (1 : N ⋊[φ] G).right = 1 := rfl | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | one_right | null |
@[simp]
inv_left (a : N ⋊[φ] G) : a⁻¹.left = φ a.right⁻¹ a.left⁻¹ := rfl
@[simp] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | inv_left | null |
inv_right (a : N ⋊[φ] G) : a⁻¹.right = a.right⁻¹ := rfl | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | inv_right | null |
inl : N →* N ⋊[φ] G where
toFun n := ⟨n, 1⟩
map_one' := rfl
map_mul' := by intros; ext <;>
simp only [mul_left, map_one, MulAut.one_apply, mul_right, mul_one]
@[simp] | def | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | inl | The canonical map `N →* N ⋊[φ] G` sending `n` to `⟨n, 1⟩` |
left_inl (n : N) : (inl n : N ⋊[φ] G).left = n := rfl
@[simp] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | left_inl | null |
right_inl (n : N) : (inl n : N ⋊[φ] G).right = 1 := rfl | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | right_inl | null |
inl_injective : Function.Injective (inl : N → N ⋊[φ] G) :=
Function.injective_iff_hasLeftInverse.2 ⟨left, left_inl⟩
@[simp] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | inl_injective | null |
inl_inj {n₁ n₂ : N} : (inl n₁ : N ⋊[φ] G) = inl n₂ ↔ n₁ = n₂ :=
inl_injective.eq_iff | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | inl_inj | null |
inr : G →* N ⋊[φ] G where
toFun g := ⟨1, g⟩
map_one' := rfl
map_mul' := by intros; ext <;> simp
@[simp] | def | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | inr | The canonical map `G →* N ⋊[φ] G` sending `g` to `⟨1, g⟩` |
left_inr (g : G) : (inr g : N ⋊[φ] G).left = 1 := rfl
@[simp] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Complement"
] | Mathlib/GroupTheory/SemidirectProduct.lean | left_inr | null |
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