Powers of Group Elements/Sum of Indices

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Theorem

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

Let $g \in G$.

Then:

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

This can also be written in additive notation as:

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

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

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

$\blacksquare$