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I often see lower bounds on the size of centralizers of involutions in finite (nonabelian) simple groups, but is there a general upper bound for the size of an involution centralizer in such a group (in function of, for instance, the size of the simple group) ? Best,

 THC
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I think the largest involution centralizers relative to $|G|$ arise in the alternating groups $A_n$. The index of the centralizer of $(1,2)(3,4)$ is about $n(n-1)(n-2)(n-3)/8 < n^4/8$ where $n < \log |G|$.

In groups of Lie type in odd characteristic, the largest involution centralizers are roughly groups of Lie type with rank one or two smaller. So in ${\rm SL}_n(q)$ for example, the largest involution centralizer, for odd $n$ and $q$, is isomorphic to ${\rm GL}_{n-1}(q)$. So, very roughly, the simple group has order about $q^{n^2-1}$ and the largest involution centralizer has order about $q^{(n-1)^2}$, so the index is something like $\exp (O((\log |G|)^{1/2}))$ which is larger as function of $|G|$ than in $A_n$.

The involution centralizers are a little different in even characteristic groups of Lie type, but again their index is roughly $q^n$ with $|G|$ about $q^{n^2}$.

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    $\begingroup$ I guess it's necessary to rely on the classification of simple groups to get such precise results, though I wonder if anything interesting can be said a priori? $\endgroup$
    – Jim Humphreys
    Commented Nov 18, 2013 at 18:02
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    $\begingroup$ If $G$ is simple and $n$ is maximal with $n!/2 \le |G|$ then, for any subgroup $H$ of $G$ we have $|G:H| \ge n$. I am suggesting that, by using CFSG and doing a lot more checking of the known structure of involutions centralizers, it could be proved that $|G:C_G(g)| \ge n(n-1)(n-2)(n-3)/8$ for any involution $g$. So perhaps the gap is not so vast! $\endgroup$
    – Derek Holt
    Commented Nov 18, 2013 at 20:38
  • $\begingroup$ Have you looked at the Brauer-Fowler theorem? en.wikipedia.org/wiki/Brauer%E2%80%93Fowler_theorem $\endgroup$
    – John Murray
    Commented Jul 13, 2014 at 18:02

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