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

Then:

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

Now let $x \in g H$.

Then:

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

Thus:

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

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

$\blacksquare$