Congruence on a group

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 $$G$$ is an equivalence relation $$\equiv$$ on $$G$$ such that:


 * $$a \equiv b \implies a^{-1} \equiv b^{-1}$$
 * $$a \equiv b, c \equiv d \implies ac \equiv bd$$

The term congruence can more generally be used for any algebra, in the theory of universal algebras.

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.