**Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.17**

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

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