I've been struggling with the following problem. I don't really know how to tackle this, but I suspect that it is not hard at all.
Let $G$ be a group and $f\colon G\to G$ an isomorphism such that $f$ is an involution ($f^2=1_G$) and $e$ is the only fixed point of $f$ (that is, $f(x)=x\implies x=e$)
I must prove that $G$ is abelian.
How should I proceed here ?
The problem gives a hint: show that every element of $G$ can be written as $x f(x)^{-1}$. I suspect that then one can use some result like "$G$ is abelian $\iff x\mapsto x^{-1}$ is a group morphism ", but I haven't tried yet because I don't see why the hint holds.
Any help is appreciated, thanks in advance!