Cosets are Equal iff Product with Inverse in Subgroup

Theorem
Let $G$ be a group and let $H$ be a subgroup of $G$.

Let $x, y \in G$.

Let:
 * $x H$ denote the left coset of $H$ by $x$;
 * $H x$ denote the right coset of $H$ by $x$.

Then:
 * $x H = y H \iff x^{-1} y \in H$;
 * $H x = H y \iff x y^{-1} \in H$.

Proof

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


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