Let $G$ be a finite group and $\alpha$ be an automorphism of $G$ which fixes only the unit of $G$ (if $\alpha(a)=a$, then $a=1$). And $\alpha^2=1$. Show that $G$ is abelian.
I think it is enough to prove $\alpha(a^{-1}b^{-1}ab)=a^{-1}b^{-1}ab$, but don't know how to get it.
