Powers of Group Elements/Product of Indices

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Theorem

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

Let $g \in G$.

Then:

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

This can also be written in additive notation as:

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

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

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

$\blacksquare$