Order of Group Element equals Order of Coprime Power/Proof 1

Proof
Let $\order g = n$.

Then from Order of Power of Group Element:
 * $\forall m \in \Z: \order {g^m} = \dfrac n {\gcd \set {m, n}}$

where $\gcd \set {m, n}$ denotes the greatest common divisor of $m$ and $n$.

Thus:
 * $\order {g^m} = \order g \iff \gcd \set {m, n} = 1$

The result follows by definition of coprime integers.