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algebra.​group.​conj

algebra.​group.​conj

source
conj_inv
conj_mul
is_conj
is_conj_iff_eq
is_conj_one_left
is_conj_one_right
is_conj_refl
is_conj_symm
is_conj_trans
monoid_hom.​map_is_conj

Conjugacy of group elements

source
def is_conj {α : Type u} [group α] :
α → α → Prop

We say that a is conjugate to b if for some c we have c * a * c⁻¹ = b.

Equations
source
theorem is_conj_refl {α : Type u} [group α] (a : α) :
is_conj a a

source
theorem is_conj_symm {α : Type u} [group α] {a b : α} :
is_conj a b → is_conj b a

source
theorem is_conj_trans {α : Type u} [group α] {a b c : α} :
is_conj a b → is_conj b c → is_conj a c

source
@[simp]
theorem is_conj_one_right {α : Type u} [group α] {a : α} :
is_conj 1 a ↔ a = 1

source
@[simp]
theorem is_conj_one_left {α : Type u} [group α] {a : α} :
is_conj a 1 ↔ a = 1

source
@[simp]
theorem conj_inv {α : Type u} [group α] {a b : α} :
(b * a * b⁻¹)⁻¹ = b * a⁻¹ * b⁻¹

source
@[simp]
theorem conj_mul {α : Type u} [group α] {a b c : α} :
b * a * b⁻¹ * (b * c * b⁻¹) = b * (a * c) * b⁻¹

source
@[simp]
theorem is_conj_iff_eq {α : Type u_1} [comm_group α] {a b : α} :
is_conj a b ↔ a = b

source
theorem monoid_hom.​map_is_conj {α : Type u} {β : Type v} [group α] [group β] (f : α →* β) {a b : α} :
is_conj a b → is_conj (⇑f a) (⇑f b)

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