# Congruence Class Modulo Subgroup is Coset

## 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 $g \bmod H$**.