Powers of Group Elements

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

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

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 $$m,n \in \mathbb{Z}$$,


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

Proof

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

From Index Laws for Semigroups: Sum of Indices, we have:

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

which translates into:

$$\forall m, n \in \mathbb{N}^*: g^m * g^n = g^{m+n}$$

From Index Laws for Monoids: Sum of Indices, this result is extended to:

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


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

From Index Laws for Semigroups: Product of Indices, we have:

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

which translates into:

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

From Index Laws for Monoids: Product of Indices, this result is extended to:

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

As all group elements are invertible by definition, these results hold for all elements of any group.

0$$.


 * Sub-case 1: Suppose $$n>0$$.  Then,

Also,


 * Sub-Case 2: Suppose $$n=0$$.  Then,

Also,

Sub-case 3: Suppose $$n<0$$.

If $$m>-n$$, then

If $$m = -n$$, then

If $$m<-n$$, then

Also,

Case 2: Let $$m=0$$ and $$n \in \mathbb{Z}$$. Then,

Also,

Case 3: Let $$m<0$$.


 * Sub-case 1: Suppose $$n>0$$.

If $$n>-m$$, then

If $$n=-m$$, then

If $$n<-m$$, then

Also,


 * Sub-Case 2: Suppose $$n=0$$.

Also,


 * Sub-Case 3: Suppose $$n<0$$.

Also,

--> QED