# Left 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 $x H$ denote the left coset of $H$ by $x$.

Then:

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

## Proof

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

$\blacksquare$