Power of Product of Commutative Elements in Group

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

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

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

This can be expressed in additive notation in the group $\struct {G, +}$ as:


 * $\forall n \in \Z: n \cdot \paren {a + b} = \paren {n \cdot a} + \paren {n \cdot b}$

Proof
By definition, all elements of a group are invertible.

Therefore the results in Power of Product of Commutative Elements in Monoid can be applied directly.