Congruence Class Modulo Subgroup is Coset

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Theorem

Let $G$ be a group.

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


Left Congruence Class

Let $\mathcal R^l_H$ be the equivalence defined as left congruence modulo $H$.

The equivalence class $\eqclass g {\mathcal R^l_H}$ of an element $g \in G$ is the left coset $g H$.


This is known as the left congruence class of $g \bmod H$.


Right Congruence Class

Let $\mathcal R^r_H$ be the equivalence defined as right congruence modulo $H$.

The equivalence class $\eqclass g {\mathcal R^r_H}$ of an element $g \in G$ is the right coset $H g$.


This is known as the right congruence class of $g \bmod H$.


Also see