Powers of Group Elements/Negative Index

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Theorem

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

Let $g \in G$.

Then:

$\forall n \in \Z: \left({g^n}\right)^{-1} = g^{-n} = \left({g^{-1}}\right)^n$

This can also be written in additive notation as:

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

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

$\forall n \in \Z: \left({g^n}\right)^{-1} = g^{-n} = \left({g^{-1}}\right)^n$

$\blacksquare$