Question 85
Let $(G, \cdot)$ be a group with no elements of order $4$, and let $f : G \rightarrow G$ be a group morphism such that $f(x) \in \{x, x^{-1}\}$, for all $x \in G$. Prove that either $f(x) = x$ for all $x \in G$, or $f(x) = x^{-1}$ for all $x \in G$.