Congruence Relation - Congruences of Groups, and Normal Subgroups and Ideals

Congruences of Groups, and Normal Subgroups and Ideals

In the particular case of groups, congruence relations can be described in elementary terms as follows: If G is a group (with identity element e and operation *) and ~ is a binary relation on G, then ~ is a congruence whenever:

1. Given any element a of G, a ~ a (reflexivity);
2. Given any elements a and b of G, if a ~ b, then b ~ a (symmetry);
3. Given any elements a, b, and c of G, if a ~ b and b ~ c, then a ~ c (transitivity);
4. Given any elements a, a', b, and b' of G, if a ~ a' and b ~ b', then a * b ~ a' * b' ;
5. Given any elements a and a' of G, if a ~ a', then a−1 ~ a' −1 (this can actually be proven from the other four, so is strictly redundant).

Conditions 1, 2, and 3 say that ~ is an equivalence relation.

A congruence ~ is determined entirely by the set {a ∈ G : a ~ e} of those elements of G that are congruent to the identity element, and this set is a normal subgroup. Specifically, a ~ b if and only if b−1 * a ~ e. So instead of talking about congruences on groups, people usually speak in terms of normal subgroups of them; in fact, every congruence corresponds uniquely to some normal subgroup of G.