Order of Group Element equals Order of Coprime Power

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

Let $g \in G$ be an element of $g$.

Let $\left\lvert{g}\right\rvert$ denote the order of $g$ in $G$.

Then:
 * $\forall m \in \Z: \left\lvert{g^m}\right\rvert = \left\lvert{g}\right\rvert \iff m \perp \left\lvert{g}\right\rvert$

where:
 * $g^m$ denotes the $m$th power of $g$ in $G$
 * $\perp$ denotes coprimality.