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 $G$:


 * $\mathcal R^l_H = \set {\tuple {x, y} \in G \times G: x^{-1} y \in H}$

This is called left congruence modulo $H$.

When $\tuple {x, y} \in \mathcal R^l_H$, we write:
 * $x \equiv^l y \pmod H$

which is read: $x$ is left congruent to $y$ 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