Definition:Congruence Modulo Subgroup/Left Congruence

Definition
Let $G$ be a group.

Let $H$ be a subgroup of $G$.

We can use $H$ to define a relation on $G$ as follows:
 * $\mathcal R^l_H := \set {\tuple {x, y} \in G \times G: x^{-1} y \in H}$

This is called left congruence modulo $H$.

Also see

 * Definition:Right Congruence Modulo Subgroup


 * Left Congruence Modulo Subgroup is Equivalence Relation
 * Definition:Left Coset
 * Definition:Left Coset Space


 * Equivalent Statements for Congruence Modulo Subgroup