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

Proof
Let $\left\lvert{g}\right\rvert = n$.

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

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

Thus:
 * $\left\lvert{g^m}\right\rvert = \left\lvert{g}\right\rvert \iff \gcd \left\{{m, n}\right\} = 1$

The result follows by definition of coprime integers.