By the famous result of Brauer and Fowler, there exist finitely many simple groups with given involution centralizer. There are many results which determine all finite simple groups with given involution centralizer. Is a complete solution known?

That is, do we have a complete classification of

  1. Groups $H$ such that $H \cong C_G(t)$ for some involution $t$ of a finite simple group $G$?
  2. For each $H$, a list of all finite simple groups $G$ such that $C_G(t) \cong H$ for some involution $t \in G$?
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    $\begingroup$ I'm pretty sure that this list is known. For simple groups of Lie type defined over a field of odd characteristic the elements are semisimple and one can use the theory of algebraic groups to help you compute the centralisers. You'll find this information in §4.5 of "The Classification of the Finite Simple Groups: Number 3" by Gorenstein--Lyons--Solomon. $\endgroup$ – Jay Taylor Dec 2 '17 at 15:48
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    $\begingroup$ For groups of lie type over a field of characteristic $p=2$ you're looking at a unipotent element, which is in bad characteristic for any group outside of type $\mathsf{A}$. However, the book by Liebeck--Seitz on unipotent conjugacy classes should contain the answer. The work by Mizuno on unipotent conjugacy classes in bad characteristic must also give this information. $\endgroup$ – Jay Taylor Dec 2 '17 at 15:50
  • $\begingroup$ For the sporadic groups, I'm certain this must be known. Brauer--Fowler is a key starting point for the classification and I'm sure one starts with the knowledge of the centralisers of involutions in sporadic groups. I would expect this to also be in the volumes by GLS. Most of the information should also be contained in the ATLAS of finite groups. $\endgroup$ – Jay Taylor Dec 2 '17 at 15:55
  • $\begingroup$ But if you mean has anyone published any lists of this type, or are there any plans to publish such lists, then I think the answer is probably no. $\endgroup$ – Derek Holt Dec 2 '17 at 16:12
  • $\begingroup$ @JayTaylor: Thanks for the comments. I knew this must be known at least in most cases, but I am less familiar with the literature. In Liebeck-Seitz for $p = 2$ it seems the centralizers of unipotent elements are given in terms of the unipotent radical of the centralizers. Maybe the precise structure of those is a bit more difficult to describe? $\endgroup$ – spin Dec 2 '17 at 16:20

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