Cosets are Equal iff Element in Other Coset

Theorem

Let $\struct {G, \circ}$ be a group.

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

Let $x, y \in G$.

Left Cosets are Equal iff Element in Other Left Coset

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

Then:

$x H = y H \iff x \in y H$

Right Cosets are Equal iff Element in Other Right Coset

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

Then:

$H x = H y \iff x \in H y$