Powers of Group Elements

Theorem

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

Let $a \in G$.

Then the following results hold:

Negative Index

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

Sum of Indices

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

Product of Indices

$\forall m, n \in \Z: \left({g^m}\right)^n = g^{m n} = \left({g^n}\right)^m$

Also see

• Exponent Combination Laws