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:

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


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