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.
$\blacksquare$