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:


 * $$x \equiv^l y \left({\bmod \, H}\right) \iff x H = y H$$
 * $$x \not \equiv^l y \left({\bmod \, H}\right) \iff x H \cap y H = \varnothing$$.

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


 * $$x \equiv^r y \left({\bmod \, H}\right) \iff H x = H y$$
 * $$x \not \equiv^r y \left({\bmod \, H}\right) \iff H x \cap H y = \varnothing$$.

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.