Congruence Class Modulo Subgroup is Coset

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

Left Congruence Class
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 $g \, \bmod \, H$.

Right Congruence Class
Similarly, 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 $g \, \bmod \, H$.

Coset Spaces form Partition
Consequently, the left coset space forms a partition of its group, and hence:

Similarly, the right coset space forms a partition of its group:

Uniqueness of Coset
Hence:
 * For each $x \in G$ there exists a unique left coset of $H$ containing $x$, that is: $x H$;
 * For each $x \in G$ there exists a unique right coset of $H$ containing $x$, that is: $H x$.

Proof of Left Congruence Class

 * 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 $g H$.

Proof of Right Congruence Class
The proof for this follows the same structure as the proof for the Left Congruence Class.

Proof that Coset Spaces form Partition
Follows directly from:


 * Congruence Modulo a Subgroup is an Equivalence;
 * Relation Partitions a Set iff Equivalence.

Proof of Uniqueness of Coset
Follows directly from:
 * Congruence Modulo a Subgroup is an Equivalence;
 * Element in its own Equivalence Class.