Powers of Group Elements

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Theorem

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

Let $a \in G$.


Then the following results hold:

Negative Index

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


Sum of Indices

$\forall m, n \in \Z: g^m \circ g^n = g^{m + n}$


Product of Indices

$\forall m, n \in \Z: \paren {g^m}^n = g^{m n} = \paren {g^n}^m$


Also see