Power of Elements is Subgroup

Theorem
Let $G$ be an abelian group.

Then for any $n \in \Z$, the set $G^n = \left\{{ x^n: x \in G }\right\}$ is a subgroup of $G$.

Moreover, if $G$ is finite, and $n$ does not divide the order of $G$, then $G^n$ is a proper subgroup of $G$.