# Powers of Group Elements

## 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

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