of A under is a substructure of B isomorphic to the quotient of A by this congruence.

On the other hand, the congruence relation induces a unique homomorphism given by
: .

Thus, there is a natural correspondence between the congruences and the homomorphisms of any given algebraic structure.

Congruences of groups, and normal subgroups and ideals

In the particular case of groups, congruence relations can be described