Definition:Congruence Modulo Subgroup

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

Left Congruence Modulo a Subgroup
Then we can use $H$ to define a relation on $G$:


 * $\mathcal R^l_H = \left\{{\left({x, y}\right) \in G \times G: x^{-1} y \in H}\right\}$

When $\left({x, y}\right) \in \mathcal R^l_H$, we write $x \equiv^l y \pmod H$.

This is called left congruence modulo $H$.

Right Congruence Modulo a Subgroup
Similarly, we can use $H$ to define another relation on $G$:


 * $\mathcal R^r_H = \left\{{\left({x, y}\right) \in G \times G: x y^{-1} \in H}\right\}$

When $\left({x, y}\right) \in \mathcal R^r_H$, we write $x \equiv^r y \pmod H$.

This is called right congruence modulo $H$.

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