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 $$x \equiv^l y \left({\bmod \, H}\right)$$.

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 $$x \equiv^r y \left({\bmod \, H}\right)$$.

This is called right congruence modulo $$H$$.

Both left congruence modulo $$H$$ and right congruence modulo $$H$$ are equivalence relations.

Also see

 * Coset
 * Coset Space


 * Equivalent Statements for Congruence Modulo a Subgroup