Exercise 1.1.17 (If $|x| = n$ then $x^{-1} = x^{n-1}$)

Question

Let $x$ be an element of $G$. Prove that if $|x| = n$ for some positive integer $n$ then $x^{-1} = x^{n-1}$.

Richard Ganaye · Accepted Answer

\begin{proof} 
If $|x| =n$, then $x^n = 1$, so 
$$x^{n-1} = x^n \cdot x^{-1} = x^{-1}.$$

\end{proof}

Answers