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} e & : n = 0 \\ a^{n-1} * a & : n > 0 \end{cases} $$

and from Index Laws for Monoids, we define:


 * $$a^n = \left({a^{-1}}\right)^{-n} : n < 0$$

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:


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