Element in Right Coset iff Product with Inverse in Subgroup

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

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

Let $x, y \in G$. Let $H \circ y$ denote the right coset of $H$ by $y$.

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

Proof
Let $\struct {G, *}$ be the opposite group of $\struct {G, \circ}$.

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

Since $H$ is closed under inverses:
 * $x \circ y^{-1} \in H \iff x^{-1} * y \in H$

By Element in Left Coset iff Product with Inverse in Subgroup:
 * $x \in y * H \iff x^{-1} * y \in H$

Hence the result.

Also see

 * Element in Left Coset iff Product with Inverse in Subgroup