Powers of Group Elements/Negative Index

Theorem
Let $\struct {G, \circ}$ be a group whose identity is $e$.

Let $g \in G$.

Then:
 * $\forall n \in \Z: \paren {g^n}^{-1} = g^{-n} = \paren {g^{-1} }^n$

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:


 * $\forall n \in \Z: \paren {g^n}^{-1} = g^{-n} = \paren {g^{-1} }^n$