# Congruence on a group

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## Definition

### Symbol-free definition

A **congruence on a group** is an equivalence relation on the elements of the group that is compatible with all the group operations.

### Definition with symbols

A congruence on a group is an equivalence relation on such that:

The term **congruence** can more generally be used for any algebra, in the theory of universal algebras. `Further information: congruence on an algebra`

## Facts

### The congruence class of the identity element

It is easy to see that the congruence class of the identity element is a normal subgroup.

Conversely, given any normal subgroup, there is a unique congruence where the congruence class of the identity element is that normal subgroup. The congruence classes here are the cosets of the normal subgroup.

### The quotient map for a congruence

Given a congruence on a group, there is a natural quotient map from the group to the set of congruence classes. Under this map, the set of congruence classes inherits a group structure. This is termed the quotient group.