More lemmas about group actions

This file contains lemmas about group actions that require more imports than Mathlib.Algebra.Group.Action.Defs offers.

def MulAction.toPerm {α : Type u_1} {β : Type u_2} [Group α] [MulAction α β] (a : α) : Equiv.Perm β

Given an action of a group α on β, each g : α defines a permutation of β.

Equations

• MulAction.toPerm a = { toFun := fun (x : β) => a • x, invFun := fun (x : β) => a⁻¹ • x, left_inv := ⋯, right_inv := ⋯ }

def AddAction.toPerm {α : Type u_1} {β : Type u_2} [AddGroup α] [AddAction α β] (a : α) : Equiv.Perm β

Given an action of an additive group α on β, each g : α defines a permutation of β.

Equations

• AddAction.toPerm a = { toFun := fun (x : β) => a +ᵥ x, invFun := fun (x : β) => -a +ᵥ x, left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem MulAction.toPerm_apply {α : Type u_1} {β : Type u_2} [Group α] [MulAction α β] (a : α) (x : β) : (MulAction.toPerm a) x = a • x

@[simp]

theorem AddAction.toPerm_apply {α : Type u_1} {β : Type u_2} [AddGroup α] [AddAction α β] (a : α) (x : β) : (AddAction.toPerm a) x = a +ᵥ x

@[simp]

theorem MulAction.toPerm_symm_apply {α : Type u_1} {β : Type u_2} [Group α] [MulAction α β] (a : α) (x : β) : (Equiv.symm (MulAction.toPerm a)) x = a⁻¹ • x

@[simp]

theorem AddAction.toPerm_symm_apply {α : Type u_1} {β : Type u_2} [AddGroup α] [AddAction α β] (a : α) (x : β) : (Equiv.symm (AddAction.toPerm a)) x = -a +ᵥ x

theorem MulAction.toPerm_injective {α : Type u_1} {β : Type u_2} [Group α] [MulAction α β] [FaithfulSMul α β] : Function.Injective MulAction.toPerm

MulAction.toPerm is injective on faithful actions.

theorem AddAction.toPerm_injective {α : Type u_1} {β : Type u_2} [AddGroup α] [AddAction α β] [FaithfulVAdd α β] : Function.Injective AddAction.toPerm

AddAction.toPerm is injective on faithful actions.

def MulAction.toPermHom (α : Type u_1) (β : Type u_2) [Group α] [MulAction α β] : α →* Equiv.Perm β

Given an action of a group α on a set β, each g : α defines a permutation of β.

Equations

• MulAction.toPermHom α β = { toFun := MulAction.toPerm, map_one' := ⋯, map_mul' := ⋯ }

@[simp]

theorem MulAction.toPermHom_apply (α : Type u_1) (β : Type u_2) [Group α] [MulAction α β] (a : α) : (MulAction.toPermHom α β) a = MulAction.toPerm a

def AddAction.toPermHom (β : Type u_2) (α : Type u_3) [AddGroup α] [AddAction α β] : α →+ Additive (Equiv.Perm β)

Given an action of an additive group α on a set β, each g : α defines a permutation of β.

Equations

• AddAction.toPermHom β α = MonoidHom.toAdditive'' (MulAction.toPermHom (Multiplicative α) β)

@[simp]

theorem AddAction.toPermHom_apply_symm_apply (β : Type u_2) (α : Type u_3) [AddGroup α] [AddAction α β] (a : α) (x : β) : (Equiv.symm ((AddAction.toPermHom β α) a)) x = (Multiplicative.ofAdd a)⁻¹ • x

@[simp]

theorem AddAction.toPermHom_apply_apply (β : Type u_2) (α : Type u_3) [AddGroup α] [AddAction α β] (a : α) (x : β) : ((AddAction.toPermHom β α) a) x = a +ᵥ x

instance Equiv.Perm.applyMulAction (α : Type u_3) : MulAction (Equiv.Perm α) α

The tautological action by Equiv.Perm α on α.

This generalizes Function.End.applyMulAction.

Equations

• Equiv.Perm.applyMulAction α = MulAction.mk ⋯ ⋯

@[simp]

theorem Equiv.Perm.smul_def {α : Type u_3} (f : Equiv.Perm α) (a : α) : f • a = f a

instance Equiv.Perm.applyFaithfulSMul (α : Type u_3) : FaithfulSMul (Equiv.Perm α) α

Equiv.Perm.applyMulAction is faithful.

Equations

• ⋯ = ⋯

theorem MulAction.bijective {α : Type u_1} {β : Type u_2} [Group α] [MulAction α β] (g : α) : Function.Bijective fun (x : β) => g • x

theorem AddAction.bijective {α : Type u_1} {β : Type u_2} [AddGroup α] [AddAction α β] (g : α) : Function.Bijective fun (x : β) => g +ᵥ x

theorem MulAction.injective {α : Type u_1} {β : Type u_2} [Group α] [MulAction α β] (g : α) : Function.Injective fun (x : β) => g • x

theorem AddAction.injective {α : Type u_1} {β : Type u_2} [AddGroup α] [AddAction α β] (g : α) : Function.Injective fun (x : β) => g +ᵥ x

theorem MulAction.surjective {α : Type u_1} {β : Type u_2} [Group α] [MulAction α β] (g : α) : Function.Surjective fun (x : β) => g • x

theorem AddAction.surjective {α : Type u_1} {β : Type u_2} [AddGroup α] [AddAction α β] (g : α) : Function.Surjective fun (x : β) => g +ᵥ x

theorem smul_left_cancel {α : Type u_1} {β : Type u_2} [Group α] [MulAction α β] (g : α) {x y : β} (h : g • x = g • y) : x = y

theorem vadd_left_cancel {α : Type u_1} {β : Type u_2} [AddGroup α] [AddAction α β] (g : α) {x y : β} (h : g +ᵥ x = g +ᵥ y) : x = y

@[simp]

theorem smul_left_cancel_iff {α : Type u_1} {β : Type u_2} [Group α] [MulAction α β] (g : α) {x y : β} : g • x = g • y ↔ x = y

@[simp]

theorem vadd_left_cancel_iff {α : Type u_1} {β : Type u_2} [AddGroup α] [AddAction α β] (g : α) {x y : β} : g +ᵥ x = g +ᵥ y ↔ x = y

theorem smul_eq_iff_eq_inv_smul {α : Type u_1} {β : Type u_2} [Group α] [MulAction α β] (g : α) {x y : β} : g • x = y ↔ x = g⁻¹ • y

theorem vadd_eq_iff_eq_neg_vadd {α : Type u_1} {β : Type u_2} [AddGroup α] [AddAction α β] (g : α) {x y : β} : g +ᵥ x = y ↔ x = -g +ᵥ y

@[simp]

theorem invOf_smul_smul {α : Type u_1} {β : Type u_2} [Monoid α] [MulAction α β] (c : α) (x : β) [Invertible c] : ⅟c • c • x = x

@[simp]

theorem smul_invOf_smul {α : Type u_1} {β : Type u_2} [Monoid α] [MulAction α β] (c : α) (x : β) [Invertible c] : c • ⅟c • x = x

theorem invOf_smul_eq_iff {α : Type u_1} {β : Type u_2} [Monoid α] [MulAction α β] {c : α} {x y : β} [Invertible c] : ⅟c • x = y ↔ x = c • y

theorem smul_eq_iff_eq_invOf_smul {α : Type u_1} {β : Type u_2} [Monoid α] [MulAction α β] {c : α} {x y : β} [Invertible c] : c • x = y ↔ x = ⅟c • y

def arrowAction {G : Type u_3} {A : Type u_4} {B : Type u_5} [DivisionMonoid G] [MulAction G A] : MulAction G (A → B)

If G acts on A, then it acts also on A → B, by (g • F) a = F (g⁻¹ • a).

Equations

• arrowAction = MulAction.mk ⋯ ⋯

def arrowAddAction {G : Type u_3} {A : Type u_4} {B : Type u_5} [SubtractionMonoid G] [AddAction G A] : AddAction G (A → B)

If G acts on A, then it acts also on A → B, by (g +ᵥ F) a = F (g⁻¹ +ᵥ a)

Equations

• arrowAddAction = AddAction.mk ⋯ ⋯

@[simp]

theorem arrowAddAction_vadd {G : Type u_3} {A : Type u_4} {B : Type u_5} [SubtractionMonoid G] [AddAction G A] (g : G) (F : A → B) (a : A) : VAdd.vadd g F a = F (-g +ᵥ a)

@[simp]

theorem arrowAction_smul {G : Type u_3} {A : Type u_4} {B : Type u_5} [DivisionMonoid G] [MulAction G A] (g : G) (F : A → B) (a : A) : SMul.smul g F a = F (g⁻¹ • a)

theorem IsUnit.smul_left_cancel {α : Type u_1} {β : Type u_2} [Monoid α] [MulAction α β] {a : α} (ha : IsUnit a) {x y : β} : a • x = a • y ↔ x = y

theorem IsAddUnit.vadd_left_cancel {α : Type u_1} {β : Type u_2} [AddMonoid α] [AddAction α β] {a : α} (ha : IsAddUnit a) {x y : β} : a +ᵥ x = a +ᵥ y ↔ x = y

@[simp]

theorem isUnit_smul_iff {α : Type u_1} {β : Type u_2} [Group α] [Monoid β] [MulAction α β] [SMulCommClass α β β] [IsScalarTower α β β] (g : α) (m : β) : IsUnit (g • m) ↔ IsUnit m