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.
(i) Reflexivity: Let then because is a subgroup of .
Hence for all . The relation is reflexive.
Hence the relation is symmetric.
Hence the relation is transitive.
Thus the relation congruence is an equivalence relation in .