# Definition:Congruence Modulo Subgroup

## Definition

Let $G$ be a group.

Let $H$ be a subgroup of $G$.

We can use $H$ to define relations on $G$ as follows:

### Left Congruence Modulo Subgroup

$\mathcal R^l_H := \set {\tuple {x, y} \in G \times G: x^{-1} y \in H}$

This is called left congruence modulo $H$.

### Right Congruence Modulo Subgroup

$\mathcal R^r_H = \set {\tuple {x, y} \in G \times G: x y^{-1} \in H}$

This is called right congruence modulo $H$.

Some authors introduce the concept of congruence modulo $H$ in the context of ring theory.

In this case, the group $G$ is taken to be the additive group of a ring.

This is acceptable, but such a treatment does presuppose that $G$ is abelian.

In such a context, all the richness of the analysis of normal subgroups is disappointingly bypassed.

## Also see

• Results about congruence modulo a subgroup can be found here.