Powers of Group Elements/Sum of Indices

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

Let $g \in G$.

Then:
 * $\forall m, n \in \Z: g^m \circ g^n = g^{m + n}$

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


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

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


 * $\forall m, n \in \Z: g^m \circ g^n = g^{m + n}$