Powers of Commutative Elements in Groups
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Theorem
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
- $a \circ b = b \circ a \iff \forall n \in \Z: \paren {a \circ b}^n = a^n \circ b^n$