Powers of Group Elements/Product of Indices

Theorem
Let $\left({G, \circ}\right)$ be a group whose identity is $e$.

Let $g \in G$.

Then:
 * $\forall m, n \in \Z: \left({g^m}\right)^n = g^{m n} = \left({g^n}\right)^m$

Additive Notation
This can also be written in additive notation as:


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

Proof
All elements of a group are invertible, so we can directly use the result from Index Laws for Monoids: Product of Indices:


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