Elements in Same Coset iff Product with Inverse in Subgroup

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

Let $$x, y \in G$$.

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

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

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

Similarly, from Congruence Class Modulo Subgroup is Coset, we have that the right cosets of $$G$$ also 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:
 * $$x \in y H$$ and $$y \in x H$$ iff $$x H = y H$$;
 * if $$x \in H y$$ and $$y \in H x$$ iff $$H x = H y$$.

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

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

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

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

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

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

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

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

Again, 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 the results:
 * $$x, y$$ are in the same left coset of $$H$$ if $$x^{-1} y \in H$$;
 * $$x, y$$ are in the same right coset of $$H$$ if $$x y^{-1} \in H$$.

Thus we have implication two ways, and the work is done.