The existence of a group automorphism with some properties implies commutativity. 
Let $G $ be a finite group, $T$ be an automorphisom of $ G $ st $ Tx = x \iff x=e $. Suppose further that $ T^2 =I $. Prove that $ G $ is abelian.

I was thinking  if  I show $ T aba^{-1} b^ {-1}=aba^ {-1}b^{-1} \forall a, b \in G$. But I was unable to show it. Please  give me any hints about it.

HINT: Consider $\sigma:x\mapsto x^{-1}T(x)$.

 Note that, in general, if $f(x)$ is a fixed point free automorphism of $G$ (fixed point free means $x=1\Leftrightarrow f(x)=x$), then $\sigma(x)=x^{-1}f(x)$ is a bijection from $G$ to $G$ (though this function is not in general a homomorphism).

