Let $G$ be an abelian group. Show that the mapping $\phi:G\to G$ given by $\phi(x) = x^{-1}$ is an automorphism of $G$.
Characterize the groups $G$ for which the map $\iota: G \to G$, sending $x \mapsto x^{-1}$ for all $x \in G$, is an automorphism of $G$