Order of Group Element equals Order of Coprime Power

Theorem
Let $\struct {G, \circ}$ be a group whose identity is $e$.

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

Let $\order g$ denote the order of $g$ in $G$.

Then:
 * $\forall m \in \Z: \order {g^m} = \order g \iff m \perp \order g$

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