# Definition:Power of Element/Group

## Definition

Let $\struct {G, \circ}$ be a group whose identity element is $e$.

Let $g \in G$.

Let $n \in \Z$.

The definition $g^n = \map {\circ^n} g$ as the $n$th power of $g$ in a monoid can be extended to allow negative values of $n$:

$g^n = \begin{cases} e & : n = 0 \\ g^{n - 1} \circ g & : n > 0 \\ \paren {g^{-n} }^{-1} & : n < 0 \end{cases}$

or

$n \cdot g = \begin{cases} e & : n = 0 \\ \paren {\paren {n - 1} \cdot g} \circ g & : n > 0 \\ -\paren {-n \cdot g} & : n < 0 \end{cases}$

The validity of this definition follows from the group axioms: $g$ has an inverse element.