Powers of Commutative Elements in Groups

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Let $\struct {G, \circ}$ be a group.

Let $a, b \in G$ such that $a$ and $b$ commute.

Then the following results hold:

Commutativity of Powers

$\forall m, n \in \Z: a^m \circ b^n = b^n \circ a^m$

Product of Commutative Elements

$a \circ b = b \circ a \iff \forall n \in \Z: \paren {a \circ b}^n = a^n \circ b^n$

Also see