Groups whose centralisers are finite

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.

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Can you give at least one motivational example for such a group? Thanks. –  Martin Brandenburg Oct 31 '11 at 10:02
I know no easy example. The only ones I could find are the so called Tarski Monsters (every proper subgroups of which -other than the identity- are cyclic of order a fixed prime p). However, I do not know how many conjugacy classes they have. –  Drike Oct 31 '11 at 11:30

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 such a group with two conjugacy classes.