Power of Product in Abelian Group

Theorem
Let $G$ be an abelian group.

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

Additive Notation
This can also be written in additive notation as:


 * $k \left({x + y}\right) = k x + k y$

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.