Documentation

Mathlib.GroupTheory.Congruence.Opposite

Congruences on the opposite of a group #

This file defines the order isomorphism between the congruences on a group G and the congruences on the opposite group Gᵒᵖ.

def Con.op {M : Type u_1} [Mul M] (c : Con M) :

If c is a multiplicative congruence on M, then (a, b) ↦ c b.unop a.unop is a multiplicative congruence on Mᵐᵒᵖ

Equations
def AddCon.op {M : Type u_1} [Add M] (c : AddCon M) :

If c is an additive congruence on M, then (a, b) ↦ c b.unop a.unop is an additive congruence on Mᵃᵒᵖ

Equations
def Con.unop {M : Type u_1} [Mul M] (c : Con Mᵐᵒᵖ) :
Con M

If c is a multiplicative congruence on Mᵐᵒᵖ, then (a, b) ↦ c bᵒᵖ aᵒᵖ is a multiplicative congruence on M

Equations
def AddCon.unop {M : Type u_1} [Add M] (c : AddCon Mᵃᵒᵖ) :

If c is an additive congruence on Mᵃᵒᵖ, then (a, b) ↦ c bᵒᵖ aᵒᵖ is an additive congruence on M

Equations

The multiplicative congruences on M bijects to the multiplicative congruences on Mᵐᵒᵖ

Equations

The additive congruences on M bijects to the additive congruences on Mᵃᵒᵖ

Equations
@[simp]
theorem Con.orderIsoOp_apply {M : Type u_1} [Mul M] (c : Con M) :
@[simp]
theorem AddCon.orderIsoOp_apply {M : Type u_1} [Add M] (c : AddCon M) :