Algebra Solutions: Basic property of inverses of group elements of finite order

Let $G$ be a group and let $x \in G$ . Prove that if $|x| = n$ for some $n \in \mathbb{Z}^+$ , then $x^{-1} = x^{n-1}$ .

We have $x \cdot x^{n-1} = x^n = 1$ , so by the uniquenes of inverses $x^{-1} = x^{n-1}$ .

Algebra Solutions