TITLE: Classification of finite simple AC groups.
QUESTION [6 upvotes]: 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.

REPLY [7 votes]: 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