The ensured existence of the inverse elements in a group allows us to extend the exponentiation we have learned for monoids by explaining what it means to raise an element of a group to a negative power.
Let \((G,\ast)\) be a group, \(x\in G\), and \(n\) an integer. We define the exponentiation to the \(n\)-th power as the binary operation "$\ast$" applied \(n\) times to the element \(x\). Like in exponentiation in a monoid, we set for all $x\in G$:
\[x^n := \begin{cases} e & \text{ if } n=0 \\ x\ast x^{n-1} & \text{ if } n > 0. \end{cases}\]
In addition, we set for negative exponents $$x^{-n}:=(x^{-1})^n.$$
In the above definition, $e\in G$ denotes the unique neutral element of $G$ and $x^{-1}$ denotes the unique inverse element of $x\in G.$
Corollaries: 1
Corollaries: 1
Definitions: 2
Explanations: 3
Proofs: 4 5 6
