Hallo!


I'm looking for a reference. I'm sure that the information I need is already in the literature but I'm having some trouble to find it. Here is the question.


Let $S$ be a non-abelian finite simple group (the only case I'm really interested in is for groups of Lie type) and let $A$ be the automorphism group of $S$. For which primes $p$, every element of order $p$ in $S$ has $C\_A(x)$ (the centralizer of $x$ in $A$) abelian? 


Typically, one might think that $p$ is a primitive prime divisor of $q^n-1$ (where $q$ is the size of the defining field of $S$ and $n$ is, roughly, its Lie rank). However, already for these elements I'm not able to find any reference for $C\_A(x)$ (despite the fact that lots is known on $C\_S(x)$).



abelian centralizers in almost simple groups
As comments by other people suggest, this kind of question requires a lot of case-by-study, even for groups of Lie type. In the latter groups, regular semisimple elements certainly get involved when the prime is different from the defining one. The regular unipotent elements in $\mathrm{PSL}\_2(\mathbb{F}\_p)$ should also be looked at. But in general irregular elements already have non-abelian centralizers in the finite simple group.


Concerning references, there are many kinds of books and papers which provide detail about the classes and centralizers in finite simple groups. Especially when there are no interesting outer automorphisms, these sources should settle your question case-by-case. Among the books is Part 3 of the ongoing AMS series by Gorenstein-Lyons-Solomon on classification of finite simple groups: they give a vast amount of detailed information about known groups, along with references. There is a lot of algebraic group literature which treats the finite groups explicitly, going back to Steinberg's Yale lectures on Chevalley groups and the detailed article by Springer and Steinberg in LN 131 based on the 1968-69 program at IAS. Much other relevant work has gone on over the years, including another AMS book to appear soon by Liebeck and Seitz *Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras*. 


I'll try to add more specific comments on your question, but I should emphasize the need to formulate it as narrowly as possible in view of the extra difficulties involved in studying sporadic groups as well as groups of Lie type for bad primes.
(Mentioning some motivation for the question might also be helpful.)


