# Definition:Congruence Modulo Subgroup

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## Contents

## 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 $H$**.

## Additive Group of Ring

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.

## Sources

- 1968: Ian D. Macdonald:
*The Theory of Groups*... (previous) ... (next): Appendix: Elementary set and number theory - 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $5$: Rings: $\S 20$. Cosets