Power of Product in Abelian Group/Additive Notation

Theorem

Let $\left({G, +}\right)$ be an abelian group.

Then:

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

Proof

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

That is:

$x + y = y + x$

The result follows from Power of Product of Commutative Elements in Group.

$\blacksquare$