Right Congruence Class Modulo Subgroup is Right Coset

Theorem
Let $G$ be a group, and let $H \le G$.

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

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

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

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

Then:

Now let $x \in g H$.

Then:

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

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

Also see

 * Left Congruence Class Modulo Subgroup is Left Coset


 * Right Coset Space forms Partition
 * Uniqueness of Cosets