Powers of Element form Subgroup

Theorem
Let $$\left({G, \circ}\right)$$ be a group.

$$\forall a \in G: H = \left\{{a^n: n \in \mathbb{Z}}\right\} \le G$$.

That is, the subset of $$G$$ comprising all elements possible as powers of $$a \in G$$ is a subgroup of $$G$$.

Proof

 * Clearly $$a \in H$$, so $$H \ne \varnothing$$.


 * Let $$x, y \in H$$.

$$ $$ $$ $$ $$

Thus by the One-step Subgroup Test, $$H \le G$$.