Commutativity of Powers in Group

Theorem

Let $\left ({G, \circ}\right)$ be a group.

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

Then:

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

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

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

Proof

By definition, all elements of a group are invertible.

Therefore Commutativity of Powers in Monoid‎ can be applied directly.

$\blacksquare$