# Relation of Congruence Modulo a Subgroup in a Group

Let be a subgroup of a group . If the element of belong to the right coset , i.e. if , i.e., if then it is said that is congruent to modulo .

Definition: Let be a subgroup of a group . For we say that is congruent to if and only if .

Symbolically, it can be expressed as if .

Theorem: The relation of congruency in a group defined by if and only if is an equivalence relation.

Proof:

(i) Reflexivity: Let then because is a subgroup of .

Hence for all . The relation is reflexive.

(ii) Symmetry:

Hence the relation is symmetric.

(iii) Transitivity:

Hence the relation is transitive.

Thus the relation congruence is an equivalence relation in .