\begin{definition}[Definition:Power of Element/Group]
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.
\end{definition}