TITLE: Groups whose centralisers are finite
QUESTION [6 upvotes]: Let $G$ be an infinite group such that the centraliser of any non central element is finite (and bounded).
Is there any structure theorem known about $G$ ?
Such a group seems to be at the other extreme of an FC-group (whose centralisers all have finite index). I can add the following requirements alltogether if need be :
1- $G$ has finitely many conjugacy classes.
2- $G$ has a trivial centre.
3- $G$ has no involution.
4- $G$ is simple.

REPLY [4 votes]: In the opposite direction, B. Hartley and M. Kuzucuoglu proved that in an infinite locally finite simple group the centralizer of every element is infinite. See Theorem A2 of the following paper:
B. Hartley - M. Kuzucuoglu: “Centralizers of Elements in Locally Finite Simple
Groups”, Proc. London Math. Soc. 62 (1991), 301-324.