Elements in Same Right Coset iff Product with Inverse in Subgroup

Theorem
Let $G$ be a group.

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

Let $x, y \in G$.

Then:
 * $x, y$ are in the same right coset of $H$ $x y^{-1} \in H$

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

First we note that, from Congruence Class Modulo Subgroup is Coset, we have that the right cosets of $G$ form a partition of $G$.

Sufficient Condition
Suppose $x, y$ are in the same left coset of $H$.

It follows from Congruence Class Modulo Subgroup is Coset that:
 * if $x \in H y$ and $y \in H x$ iff $H x = H y$

From Cosets are Equal iff Product with Inverse in Subgroup, we have that:
 * $H x = H y \iff x y^{-1} \in H$

So if $x, y$ are in the same right coset of $H$ then $x y^{-1} \in H$.

Necessary Condition
Now suppose that $x y^{-1} \in H$.

From Right Cosets are Equal iff Product with Inverse in Subgroup, we have that:
 * $H x = H y \iff x y^{-1} \in H$

It follows from Congruence Class Modulo Subgroup is Coset that:
 * $x \in H y$ and $y \in H x$ iff $H x = H y$

and so:
 * $x, y$ are in the same left coset of $H$

Hence $x, y$ are in the same right coset of $H$ if $x y^{-1} \in H$.

Also see

 * Elements in Same Left Coset iff Product with Inverse in Subgroup