Powers of Commutative Elements in Groups

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

Let $a, b \in G$.

Then the following results hold:


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

This can be expressed in additive notation in the abelian group $\left ({G, +}\right)$ as follows:


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

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.