Powers of Group Elements

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

Let $a \in G$, and let $n \in \Z$.

From Power of an Element, we define:


 * $a^n =

\begin{cases} a & : n = 1 \\ a^{n-1} * a & : n > 1 \end{cases} $

and from Index Laws for Monoids, we define:


 * $a^n =

\begin{cases} e & : n = 0 \\ \left({a^{-1}}\right)^{-n} & : n < 0 \end{cases} $

Theorem
For any element $g$ in a group $G$ and $m, n \in \Z$,


 * $g^m * g^n = g^{m + n}$;
 * $\left({g^m}\right)^n = g^{m n} = \left({g^n}\right)^m$.

Proof

 * $g^m * g^n = g^{m + n}$:

All elements of a group are invertible, so we can directly use the result from Index Laws for Monoids: Sum of Indices:


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


 * $\left({g^m}\right)^n = g^{m n}$:

All elements of a group are invertible, so we can directly use the result from Index Laws for Monoids: Product of Indices:


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