Cosets are Equal iff Product with Inverse in Subgroup

Theorem

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

Let $x, y \in G$.

Left Cosets are Equal iff Product with Inverse in Subgroup

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

Then:

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

Right Cosets are Equal iff Product with Inverse in Subgroup

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

Then:

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