Powers of Group Elements/Negative Index

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

Let $g \in G$.

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

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


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

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


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