Left Congruence Class Modulo Subgroup is Left Coset

Theorem

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

Let $\RR^l_H$ be the equivalence defined as left congruence modulo $H$.

The equivalence class $\eqclass g {\RR^l_H}$ 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 $x \in \eqclass g {\RR^l_H}$.

Then:

 $\ds x$ $\in$ $\ds \eqclass g {\RR^l_H}$ $\ds \leadsto \ \$ $\ds \exists h \in H: \,$ $\ds g^{-1} x$ $=$ $\ds h$ Definition of Left Congruence Modulo $H$ $\ds \leadsto \ \$ $\ds \exists h \in H: \,$ $\ds x$ $=$ $\ds g h$ Group properties $\ds \leadsto \ \$ $\ds x$ $\in$ $\ds g H$ Definition of Left Coset $\ds \leadsto \ \$ $\ds \eqclass g {\RR^l_H}$ $\subseteq$ $\ds g H$ Definition of Subset

Now let $x \in g H$.

Then:

 $\ds x$ $\in$ $\ds g H$ $\ds \leadsto \ \$ $\ds \exists h \in H: \,$ $\ds x$ $=$ $\ds g h$ Definition of Left Coset $\ds \leadsto \ \$ $\ds g^{-1} x$ $=$ $\ds h \in H$ Group properties $\ds \leadsto \ \$ $\ds x$ $\in$ $\ds \eqclass g {\RR^l_H}$ Definition of Left Congruence Modulo $H$ $\ds \leadsto \ \$ $\ds$ $=$ $\ds g H \subseteq \eqclass g {\RR^l_H}$ Definition of Subset

Thus:

$\eqclass g {\RR^l_H} = g H$

that is, the equivalence class $\eqclass g {\RR^l_H}$ of an element $g \in G$ equals the left coset $g H$.

$\blacksquare$