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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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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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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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