Powers of Commutative Elements in Groups

Theorem
Let $$\left ({S, \circ}\right)$$ be an abelian group.

Then the following results hold:


 * $$\forall m, n \in \mathbb{Z}: a^m \circ b^n = b^n \circ a^m$$
 * $$\forall n \in \mathbb{Z}: \left({a \circ b}\right)^n = a^n \circ b^n$$

Proof
By definition:
 * all elements of a group are invertible;
 * all elements of an abelian group are commutative with each other.

Therefore the results in Powers of Commutative Elements in Monoids‎ can be applied directly.