Left Congruence Class Modulo Subgroup is Left Coset

Theorem
Let $G$ be a group, and let $H \le G$. Let $\mathcal R^l_H$ be the equivalence defined as left congruence modulo $H$.

The equivalence class $\left[\!\left[{g}\right]\!\right]_{\mathcal R^l_H}$ of an element $g \in G$ is the left coset $g H$.

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

Proof
Let $x \in \left[\!\left[{g}\right]\!\right]_{\mathcal R^l_H}$.

Then:

Now let $x \in g H$.

Then:

Thus:
 * $\left[\!\left[{g}\right]\!\right]_{\mathcal R^l_H} = g H$

that is, the equivalence class $\left[\!\left[{g}\right]\!\right]_{\mathcal R^l_H}$ of an element $g \in G$ equals the left coset $g H$.

Also see

 * Right Congruence Class Modulo Subgroup is Coset


 * Left Coset Space forms Partition
 * Uniqueness of Cosets