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.



Groups whose centralisers are finite
Free Burnside groups of sufficiently large odd exponents $p$ ($p\ge 665$) have all centralizers of nontrivial elements cyclic of order $p$ by a result of Adian. By a result of S. Ivanov, there are groups with this property and finite number ($p$) of conjugacy classes, provided $p$ is big enough and odd. All these (and many other similar) results can be found in Olshansky's book "Geometry of defining relations in groups". Osin [constructed](https://arxiv.org/abs/math/0411039) such a group with two conjugacy classes.


