Cosets in Abelian Group

Theorem
Let $G$ be an abelian group.

Then every right coset modulo $H$ is a left coset modulo $H$.

That is:
 * $\forall x \in G: x H = H x$

In an abelian group, therefore, we can talk about congruence modulo $H$ and not worry about whether it is left or right.