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

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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.


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.

$\blacksquare$