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:

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


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