I’m trying to work through a practice problem but I don’t have the solutions the problems statement is as follows:
An isomorphism from an object to itself is called an automorphism. Let $\phi :G\to G$ be defined by $\phi(g)=g^{-1}$. Then $\phi$ is an automorphism of $G$.
My gut instinct is that this is false, for we could have an object $G$ that doesn’t contain inverses. But my only concern is this valid or are automorphisms only defined on groups, rings, etc.
Thanks!
Even if $G$ is a group (in which case each element has an inverse) this is false in general. Actually, it is not hard to prove that $\phi$ is an automorphism if and only if $G$ is Abelian.