Powers of Group Elements/Negative Index

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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:

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


Proof

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

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

$\blacksquare$


Sources