# Right Cosets are Equal iff Product with Inverse in Subgroup

## Theorem

Let $G$ be a group.

Let $H$ be a subgroup of $G$.

Let $x, y \in G$.

Let $H x$ denote the right coset of $H$ by $x$.

Then:

$H x = H y \iff x y^{-1} \in H$

## Proof

 $\displaystyle H x$ $=$ $\displaystyle H y$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle x$ $\equiv^r$ $\displaystyle y \bmod H$ Right Coset Space forms Partition $\displaystyle \leadstoandfrom \ \$ $\displaystyle x y^{-1}$ $\in$ $\displaystyle H$ Equivalent Statements for Congruence Modulo Subgroup

$\blacksquare$