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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 $C_S(x)$).

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  • $\begingroup$ You probably know this, but the non-abelian finite simple groups typically have a very small outer automorphism group (which can be described explicitly), see e.g. en.wikipedia.org/wiki/List_of_finite_simple_groups, so the group $A$ is just a little bit bigger than $S$. Wouldn't you be able to deduce information about $C_A(x)$ from $C_S(x)$ without too much effort? $\endgroup$
    – Tom De Medts
    Commented Dec 15, 2011 at 9:42
  • $\begingroup$ Another comment: a classical reference for the description of the automorphism groups of the classical groups of Lie type, is J. Dieudonné, "La Géométrie des groupes classiques", chapter IV. $\endgroup$
    – Tom De Medts
    Commented Dec 15, 2011 at 9:45
  • $\begingroup$ Hi, yes, I'm very much aware of the structure of $Out(S)$. Up to innerdiagonal automorphisms and field automorphisms it is fairly easy (as you say) to control $C_A(x)$. I'm not 100% certain of what happens when I use graph automorphisms...and here some work is required. My question looks so natural that I was hoping that has been already considered. Thanks for your comment! $\endgroup$
    – user19977
    Commented Dec 15, 2011 at 9:46
  • $\begingroup$ @unknown: I've edited the list of tags to make better contact with some of the areas actually involved. Feel free to edit further. $\endgroup$
    – Jim Humphreys
    Commented Dec 15, 2011 at 15:27

2 Answers 2

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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.)

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  • $\begingroup$ Dear All, thank you very much for your answers. So far I had a quick look at the references you suggested and I've realized that (as you mention) there is a lot of work to do for a complete answer to my "innocent looking" question. Considering the problem I was working on, I see that there is no point in pursuing this direction and I think that I will move to a different problem. $\endgroup$
    – user19977
    Commented Dec 17, 2011 at 9:16
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If the group is $(P)SL_n(q)$ then you need a prime p such that the element is regular semisimple. One can achieve this by forcing $x$ to be in the non-split torus, i.e., by taking a prime $p$ such that $p|q^n-1$ and does not divide $q^{k}-1$ for any $k < n$. (these are called Zsigmondy primes, and they exist for almost all pairs of $n,q$)

If $n \geq 3$ you can also take Zsigmondy primes for $n-1,q$ because order $p$ will force the element to be in the nosplit torus of $SL_{n-1}$.

Finally for $n=2$ you can take $p$ to be the defining characteristic.

I do not know what is going on for other classical groups :(

For the alternating groups you can only take $p=n,n-1,n-2$ (if any of these are primes)

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