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A group is called an AC-group if the centralizer of every non-central element is abelian. So far i have known only one class of groups which is a finite AC simple group, namely the group $PSL(2,q)$, where $q \equiv 0 \operatorname{ mod } 4$. Are there any other finite simple groups which are AC-groups? Thanks in advance.

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It was proved in the 1950s by Fowler, Suzuki and Wall, that the only simple CA-groups are the ones you mention, ${\rm PSL}(2,2^n)$ with $n \ge 2$. In fact nonsolvable CN-groups (centralizers of all nonidentity elements are nilpotent) were classified by Suzuki in his paper

Finite Groups with Nilpotent Centralizers, Michio Suzuki, Transactions of the American Mathematical Society Vol. 99, No. 3 (Jun., 1961), pp. 425-470

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  • $\begingroup$ I read in a paper of Prof. G. Glauberman, that the result was proved by Brauer, Suzuki and Wall, the result was obtained independently by some of all the three authors in 1953 or 1954, and it is published in the joint paper projecteuclid.org/DPubS/Repository/1.0/…), $\endgroup$ – Yassine Guerboussa Sep 26 '13 at 19:32

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