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 $m, n \in \Z$,


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

Additive Notation
This can also be written in additive notation as:


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

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:


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