mathlib documentation

data.equiv.mul_add_aut

Multiplicative and additive group automorphisms #

This file defines the automorphism group structure on add_aut R := add_equiv R R and mul_aut R := mul_equiv R R.

Implementation notes #

The definition of multiplication in the automorphism groups agrees with function composition, multiplication in equiv.perm, and multiplication in category_theory.End, but not with category_theory.comp.

This file is kept separate from data/equiv/mul_add so that group_theory.perm is free to use equivalences (and other files that use them) before the group structure is defined.

Tags #

mul_aut, add_aut

def mul_aut (M : Type u_1) [has_mul M] :
Type u_1

The group of multiplicative automorphisms.

Equations
def add_aut (M : Type u_1) [has_add M] :
Type u_1

The group of additive automorphisms.

Equations
@[instance]
def mul_aut.group (M : Type u_2) [has_mul M] :

The group operation on multiplicative automorphisms is defined by λ g h, mul_equiv.trans h g. This means that multiplication agrees with composition, (g*h)(x) = g (h x).

Equations
@[instance]
def mul_aut.inhabited (M : Type u_2) [has_mul M] :
Equations
@[simp]
theorem mul_aut.coe_mul (M : Type u_2) [has_mul M] (e₁ e₂ : mul_aut M) :
e₁ * e₂ = e₁ e₂
@[simp]
theorem mul_aut.coe_one (M : Type u_2) [has_mul M] :
theorem mul_aut.mul_def (M : Type u_2) [has_mul M] (e₁ e₂ : mul_aut M) :
e₁ * e₂ = mul_equiv.trans e₂ e₁
theorem mul_aut.one_def (M : Type u_2) [has_mul M] :
theorem mul_aut.inv_def (M : Type u_2) [has_mul M] (e₁ : mul_aut M) :
@[simp]
theorem mul_aut.mul_apply (M : Type u_2) [has_mul M] (e₁ e₂ : mul_aut M) (m : M) :
(e₁ * e₂) m = e₁ (e₂ m)
@[simp]
theorem mul_aut.one_apply (M : Type u_2) [has_mul M] (m : M) :
1 m = m
@[simp]
theorem mul_aut.apply_inv_self (M : Type u_2) [has_mul M] (e : mul_aut M) (m : M) :
e (e⁻¹ m) = m
@[simp]
theorem mul_aut.inv_apply_self (M : Type u_2) [has_mul M] (e : mul_aut M) (m : M) :
e⁻¹ (e m) = m
def mul_aut.to_perm (M : Type u_2) [has_mul M] :

Monoid hom from the group of multiplicative automorphisms to the group of permutations.

Equations
@[instance]

The tautological action by mul_aut M on M.

This generalizes function.End.apply_mul_action.

Equations
@[simp]
theorem mul_aut.smul_def {M : Type u_1} [monoid M] (f : mul_aut M) (a : M) :
f a = f a
@[instance]

mul_aut.apply_mul_action is faithful.

def mul_aut.conj {G : Type u_3} [group G] :

Group conjugation, mul_aut.conj g h = g * h * g⁻¹, as a monoid homomorphism mapping multiplication in G into multiplication in the automorphism group mul_aut G. See also the type conj_act G for any group G, which has a mul_action (conj_act G) G instance where conj G acts on G by conjugation.

Equations
@[simp]
theorem mul_aut.conj_apply {G : Type u_3} [group G] (g h : G) :
@[simp]
theorem mul_aut.conj_symm_apply {G : Type u_3} [group G] (g h : G) :
@[simp]
theorem mul_aut.conj_inv_apply {G : Type u_3} [group G] (g h : G) :
@[instance]
def add_aut.group (A : Type u_1) [has_add A] :

The group operation on additive automorphisms is defined by λ g h, add_equiv.trans h g. This means that multiplication agrees with composition, (g*h)(x) = g (h x).

Equations
@[instance]
def add_aut.inhabited (A : Type u_1) [has_add A] :
Equations
@[simp]
theorem add_aut.coe_mul (A : Type u_1) [has_add A] (e₁ e₂ : add_aut A) :
e₁ * e₂ = e₁ e₂
@[simp]
theorem add_aut.coe_one (A : Type u_1) [has_add A] :
theorem add_aut.mul_def (A : Type u_1) [has_add A] (e₁ e₂ : add_aut A) :
e₁ * e₂ = add_equiv.trans e₂ e₁
theorem add_aut.one_def (A : Type u_1) [has_add A] :
theorem add_aut.inv_def (A : Type u_1) [has_add A] (e₁ : add_aut A) :
@[simp]
theorem add_aut.mul_apply (A : Type u_1) [has_add A] (e₁ e₂ : add_aut A) (a : A) :
(e₁ * e₂) a = e₁ (e₂ a)
@[simp]
theorem add_aut.one_apply (A : Type u_1) [has_add A] (a : A) :
1 a = a
@[simp]
theorem add_aut.apply_inv_self (A : Type u_1) [has_add A] (e : add_aut A) (a : A) :
e⁻¹ (e a) = a
@[simp]
theorem add_aut.inv_apply_self (A : Type u_1) [has_add A] (e : add_aut A) (a : A) :
e (e⁻¹ a) = a
def add_aut.to_perm (A : Type u_1) [has_add A] :

Monoid hom from the group of multiplicative automorphisms to the group of permutations.

Equations
@[instance]

The tautological action by add_aut A on A.

This generalizes function.End.apply_mul_action.

Equations
@[simp]
theorem add_aut.smul_def {A : Type u_1} [add_monoid A] (f : add_aut A) (a : A) :
f a = f a
def add_aut.conj {G : Type u_3} [add_group G] :

Additive group conjugation, add_aut.conj g h = g + h - g, as an additive monoid homomorphism mapping addition in G into multiplication in the automorphism group add_aut G (written additively in order to define the map).

Equations
@[simp]
theorem add_aut.conj_apply {G : Type u_3} [add_group G] (g h : G) :
(add_aut.conj g) h = g + h - g
@[simp]
theorem add_aut.conj_symm_apply {G : Type u_3} [add_group G] (g h : G) :
@[simp]
theorem add_aut.conj_inv_apply {G : Type u_3} [add_group G] (g h : G) :