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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
up vote 12 down vote accepted

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.

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@Mark Sapir & Salvo Thank you for these precise answers and references ! – Drike Nov 1 '11 at 7:24
Thank you very much for all this information! From Osin's paper, I understand that there is a non-trivial finitely generated group with two conjugacy classes. How can I see (argument or reference) that we can ask for the additional "finite centraliser" condition to hold? – user56097 Mar 9 at 14:39

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.

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