Definition:Congruence Modulo Subgroup

Definition
Let $G$ be a group, and let $H$ be a subgroup of $G$.

Right Congruence Modulo a Subgroup
Both left congruence modulo $H$ and right congruence modulo $H$ are equivalence relations.

Alternative Treatment
Some authors introduce this concept in the context of ring theory, in which the group $G$ is taken to be the additive group of a ring. This is acceptable, but such a treatment does presuppose that the group in question is abelian. Thus all the richness of the analysis of normal subgroups is disappointingly bypassed.

Also see

 * Coset
 * Coset Space


 * Equivalent Statements for Congruence Modulo a Subgroup