Suppose \(G\) is a group and \(H\subseteq G\) a subgroup . The relation congruence modulo \(H\) is defined on \(G\) by declaring \(g\equiv g' \,(\mod H)\) if and only if \(g^{-1}g'\in H\).
Remarks
- Congruence modulo \(H\) is an equivalence relation .
- The equivalence classes of congruence module \(H\) are the left cosets of \(H\) .