# Classification of finite simple AC groups.

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.

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