# 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.

## Proof

 $\displaystyle$  $\displaystyle \forall x, y \in G: x^{-1} y = y x^{-1}$ $\displaystyle$ $\leadsto$ $\displaystyle \paren {x \equiv^l y \pmod H \iff y \equiv^r x \pmod H}$ Definition of Congruence Modulo Subgroup $\displaystyle$ $\leadsto$ $\displaystyle \paren {x \equiv^l y \pmod H \iff x \equiv^r y \pmod H}$ Congruence Modulo Subgroup is Equivalence Relation, therefore Symmetric

$\blacksquare$