Definition:Power of Element/Group

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

Let $g \in G$.

Let $n \in \Z$.

The definition $g^n = \circ ^n \left({g}\right)$ 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 \\ \left({g^{-n}}\right)^{-1} & : n < 0 \end{cases}$

or


 * $n \cdot g = \begin{cases}

e & : n = 0 \\ \left({\left({n - 1}\right) \cdot g}\right) \circ g & : n > 0 \\ - \left({-n \cdot g}\right) & : n < 0 \end{cases}$

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