Left Congruence Class Modulo Subgroup is Left Coset

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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 $\eqclass g {\mathcal R^l_H}$ 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 $x \in \eqclass g {\mathcal R^l_H}$.

Then:

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

Now let $x \in g H$.

Then:

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

Thus:

$\eqclass g {\mathcal R^l_H} = g H$

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

$\blacksquare$