Definition:Congruence Modulo Subgroup/Left Congruence

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

Left congruence modulo $H$ is an equivalence relation.

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

 * Definition:Right Congruence Modulo Subgroup


 * Definition:Left Coset
 * Definition:Left Coset Space


 * Equivalent Statements for Congruence Modulo Subgroup