Powers of Group Elements/Product of Indices/Additive Notation

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

Let $g \in G$.

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

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


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

where in this context:
 * the group product operator is $+$
 * the $n$th power of $g$ is denoted $n g$