As far as I have heard, centralizers play an important role in the theory of groups. My question arises from curiosity and the desire to understand how much control centralizers have over the group. Here we go:
If $G$ is group such that centralizer of every element (except the identity) of $G$ is finite, then is $G$ necessarily a finite group?
If the answer is "No", then such an infinite group will be heavily non-abelian (intuitively). Well, for one thing, $Z(G)=1$. Also since every element commutes with its own powers, it looks like every element would have to have a finite order. So my question seems to be stronger version of this.
I appreciate any pointers :)