# Right Congruence Class Modulo Subgroup is Right Coset

## Theorem

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

Let $\RR^r_H$ be the equivalence defined as right congruence modulo $H$.

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

Then:

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

Now let $x \in g H$.

Then:

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

Thus:

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

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

$\blacksquare$