Group actions applied to various types of group

This file contains lemmas about SMul on GroupWithZero, and Group.

instance CancelMonoidWithZero.faithfulSMul {α : Type u} [CancelMonoidWithZero α] [Nontrivial α] : FaithfulSMul α α

Monoid.toMulAction is faithful on nontrivial cancellative monoids with zero.

Equations

• ⋯ = ⋯

@[simp]

theorem inv_smul_smul₀ {α : Type u} {β : Type v} [GroupWithZero α] [MulAction α β] {c : α} (hc : c ≠ 0) (x : β) : c⁻¹ • c • x = x

@[simp]

theorem smul_inv_smul₀ {α : Type u} {β : Type v} [GroupWithZero α] [MulAction α β] {c : α} (hc : c ≠ 0) (x : β) : c • c⁻¹ • x = x

theorem inv_smul_eq_iff₀ {α : Type u} {β : Type v} [GroupWithZero α] [MulAction α β] {a : α} (ha : a ≠ 0) {x : β} {y : β} : a⁻¹ • x = y ↔ x = a • y

theorem eq_inv_smul_iff₀ {α : Type u} {β : Type v} [GroupWithZero α] [MulAction α β] {a : α} (ha : a ≠ 0) {x : β} {y : β} : x = a⁻¹ • y ↔ a • x = y

@[simp]

theorem Commute.smul_right_iff₀ {α : Type u} {β : Type v} [GroupWithZero α] [MulAction α β] [Mul β] [SMulCommClass α β β] [IsScalarTower α β β] {a : β} {b : β} {c : α} (hc : c ≠ 0) : Commute a (c • b) ↔ Commute a b

@[simp]

theorem Commute.smul_left_iff₀ {α : Type u} {β : Type v} [GroupWithZero α] [MulAction α β] [Mul β] [SMulCommClass α β β] [IsScalarTower α β β] {a : β} {b : β} {c : α} (hc : c ≠ 0) : Commute (c • a) b ↔ Commute a b

@[simp]

theorem Equiv.smulRight_apply {α : Type u} {β : Type v} [GroupWithZero α] [MulAction α β] {a : α} (ha : a ≠ 0) (b : β) : (Equiv.smulRight ha) b = a • b

@[simp]

theorem Equiv.smulRight_symm_apply {α : Type u} {β : Type v} [GroupWithZero α] [MulAction α β] {a : α} (ha : a ≠ 0) (b : β) : (Equiv.smulRight ha).symm b = a⁻¹ • b

def Equiv.smulRight {α : Type u} {β : Type v} [GroupWithZero α] [MulAction α β] {a : α} (ha : a ≠ 0) : β ≃ β

Right scalar multiplication as an order isomorphism.

Equations

• Equiv.smulRight ha = { toFun := fun (b : β) => a • b, invFun := fun (b : β) => a⁻¹ • b, left_inv := ⋯, right_inv := ⋯ }

theorem MulAction.bijective₀ {α : Type u} {β : Type v} [GroupWithZero α] [MulAction α β] {a : α} (ha : a ≠ 0) : Function.Bijective fun (x : β) => a • x

theorem MulAction.injective₀ {α : Type u} {β : Type v} [GroupWithZero α] [MulAction α β] {a : α} (ha : a ≠ 0) : Function.Injective fun (x : β) => a • x

theorem MulAction.surjective₀ {α : Type u} {β : Type v} [GroupWithZero α] [MulAction α β] {a : α} (ha : a ≠ 0) : Function.Surjective fun (x : β) => a • x

@[simp]

theorem DistribMulAction.toAddEquiv_apply {α : Type u} (β : Type v) [Group α] [AddMonoid β] [DistribMulAction α β] (x : α) : ∀ (a : β), (DistribMulAction.toAddEquiv β x) a = x • a

@[simp]

theorem DistribMulAction.toAddEquiv_symm_apply {α : Type u} (β : Type v) [Group α] [AddMonoid β] [DistribMulAction α β] (x : α) : ∀ (a : β), (DistribMulAction.toAddEquiv β x).symm a = x⁻¹ • a

def DistribMulAction.toAddEquiv {α : Type u} (β : Type v) [Group α] [AddMonoid β] [DistribMulAction α β] (x : α) : β ≃+ β

Each element of the group defines an additive monoid isomorphism.

This is a stronger version of MulAction.toPerm.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem DistribMulAction.toAddAut_apply (α : Type u) (β : Type v) [Group α] [AddMonoid β] [DistribMulAction α β] (x : α) : (DistribMulAction.toAddAut α β) x = DistribMulAction.toAddEquiv β x

def DistribMulAction.toAddAut (α : Type u) (β : Type v) [Group α] [AddMonoid β] [DistribMulAction α β] : α →* AddAut β

Each element of the group defines an additive monoid isomorphism.

This is a stronger version of MulAction.toPermHom.

Equations

• DistribMulAction.toAddAut α β = { toFun := DistribMulAction.toAddEquiv β, map_one' := ⋯, map_mul' := ⋯ }

def DistribMulAction.toAddEquiv₀ {α : Type u_1} (β : Type u_2) [GroupWithZero α] [AddMonoid β] [DistribMulAction α β] (x : α) (hx : x ≠ 0) : β ≃+ β

Each non-zero element of a GroupWithZero defines an additive monoid isomorphism of an AddMonoid on which it acts distributively. This is a stronger version of DistribMulAction.toAddMonoidHom.

Equations

• DistribMulAction.toAddEquiv₀ β x hx = let __src := DistribMulAction.toAddMonoidHom β x; { toFun := (↑__src).toFun, invFun := fun (b : β) => x⁻¹ • b, left_inv := ⋯, right_inv := ⋯, map_add' := ⋯ }

theorem smul_eq_zero_iff_eq {α : Type u} {β : Type v} [Group α] [AddMonoid β] [DistribMulAction α β] (a : α) {x : β} : a • x = 0 ↔ x = 0

theorem smul_ne_zero_iff_ne {α : Type u} {β : Type v} [Group α] [AddMonoid β] [DistribMulAction α β] (a : α) {x : β} : a • x ≠ 0 ↔ x ≠ 0

@[simp]

theorem MulDistribMulAction.toMulEquiv_symm_apply {α : Type u} (β : Type v) [Group α] [Monoid β] [MulDistribMulAction α β] (x : α) : ∀ (a : β), (MulDistribMulAction.toMulEquiv β x).symm a = x⁻¹ • a

@[simp]

theorem MulDistribMulAction.toMulEquiv_apply {α : Type u} (β : Type v) [Group α] [Monoid β] [MulDistribMulAction α β] (x : α) : ∀ (a : β), (MulDistribMulAction.toMulEquiv β x) a = x • a

def MulDistribMulAction.toMulEquiv {α : Type u} (β : Type v) [Group α] [Monoid β] [MulDistribMulAction α β] (x : α) : β ≃* β

Each element of the group defines a multiplicative monoid isomorphism.

This is a stronger version of MulAction.toPerm.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem MulDistribMulAction.toMulAut_apply (α : Type u) (β : Type v) [Group α] [Monoid β] [MulDistribMulAction α β] (x : α) : (MulDistribMulAction.toMulAut α β) x = MulDistribMulAction.toMulEquiv β x

def MulDistribMulAction.toMulAut (α : Type u) (β : Type v) [Group α] [Monoid β] [MulDistribMulAction α β] : α →* MulAut β

Each element of the group defines a multiplicative monoid isomorphism.

This is a stronger version of MulAction.toPermHom.

Equations

• MulDistribMulAction.toMulAut α β = { toFun := MulDistribMulAction.toMulEquiv β, map_one' := ⋯, map_mul' := ⋯ }

def arrowMulDistribMulAction {G : Type u_1} {A : Type u_2} {B : Type u_3} [Group G] [MulAction G A] [Monoid B] : MulDistribMulAction G (A → B)

When B is a monoid, ArrowAction is additionally a MulDistribMulAction.

Equations

• arrowMulDistribMulAction = MulDistribMulAction.mk ⋯ ⋯

@[simp]

theorem mulAutArrow_apply_symm_apply {G : Type u_1} {A : Type u_2} {H : Type u_3} [Group G] [MulAction G A] [Monoid H] (x : G) : ∀ (a : A → H) (a_1 : A), (MulEquiv.symm (mulAutArrow x)) a a_1 = (x⁻¹ • a) a_1

@[simp]

theorem mulAutArrow_apply_apply {G : Type u_1} {A : Type u_2} {H : Type u_3} [Group G] [MulAction G A] [Monoid H] (x : G) : ∀ (a : A → H) (a_1 : A), (mulAutArrow x) a a_1 = (x • a) a_1

def mulAutArrow {G : Type u_1} {A : Type u_2} {H : Type u_3} [Group G] [MulAction G A] [Monoid H] : G →* MulAut (A → H)

Given groups G H with G acting on A, G acts by multiplicative automorphisms on A → H.

Equations

• mulAutArrow = MulDistribMulAction.toMulAut G (A → H)

@[simp]

theorem IsUnit.smul_eq_zero {α : Type u} {β : Type v} [Monoid α] [AddMonoid β] [DistribMulAction α β] {u : α} (hu : IsUnit u) {x : β} : u • x = 0 ↔ x = 0

theorem IsUnit.smul_sub_iff_sub_inv_smul {α : Type u} {β : Type v} [Group α] [Monoid β] [AddGroup β] [DistribMulAction α β] [IsScalarTower α β β] [SMulCommClass α β β] (r : α) (a : β) : IsUnit (r • 1 - a) ↔ IsUnit (1 - r⁻¹ • a)