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:
 * $(1) \quad G$ is finite

and:
 * $(2) \quad n \ne 1$ is a divisor of the order of $G$

then $G^n$ is a proper subgroup of $G$.

Proof
Obviously, the set $G^n$ is not empty. So let $x^n, y^n \in G^n$.

We compute as follows:
 * $x^n\left({y^n}\right)^{-1} = x^n \left({y^{-1}}\right)^n = \left({xy^{-1}}\right)^n$

Here, the latter step follows from Powers of Product in Abelian Group.

It follows that $x^n\left({y^n}\right)^{-1} \in G^n$.

From the One-step Subgroup Test, we conclude that $G^n$ is a subgroup of $G$.

Now let $n \ne 1$ such that $n \backslash \left|{G}\right|$.

For $G^n$ to be a proper subgroup, it is enough to show that the map $x\mapsto x^n$ is not injective.

Now as $n \ne 1$, find a prime $p$ such that $p \backslash n$.

By Lagrange's theorem, there is an element $x$ with order $p$.

It then follows that $x^n = e$, but $x \ne e$.

Therefore, the map $x \mapsto x^n$ is not injective, hence the result.