Power of Product in Abelian Group

Theorem
Let $$G$$ be an abelian group.

Then:
 * $$\forall x, y \in G: \forall k \in \Z: \left({xy}\right)^k = x^k y^k$$

Proof
By definition of abelian group, $$x$$ and $$y$$ commute.

That is:
 * $$xy = yx$$

The result follows from Powers of Commutative Elements in Monoids: Product of Commutative Elements.