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Let $G$ be a finite simple group of Lie type over a finite field ($F_q$) of order $q$ with characteristic $p\neq 2$. Suppose $S$ is $2$-sylow subgroup of $G$ and is not abelian. I have two question which the first one includes the second.

  1. Is it possible that centralizer of an involution in $G$ be a $2$-group?
  2. Is it possible that centralizer of a central involution of $S$ in $G$ be a $2$-group?
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  • $\begingroup$ So it seems be true whenever the characteristic is an odd number. $\endgroup$ – Adeleh May 6 '14 at 18:08
  • $\begingroup$ @ Jack Schmidt: yes! I had a mistake! Do you know any reference that introduces the centralizer of involutions in finite simple groups directly other than "The classification of the finite simple groups, number 3"? $\endgroup$ – Adeleh May 6 '14 at 18:33
  • $\begingroup$ No. The other references are less complete or less clear. Typically though, the centralizers are huge. GL(n,K) for an algebraically closed field of characteristic not 2 is a good example group to understand. Groups of Lie type are very similar, and finite fields just complicate things (and GL(n,K) is already pretty crazy). In GL: an element of order 2 is conjugate to a diagonal element with ±1 on the diagonal. Its centralizer is GL(m,K) x GL(n-m,K) where m is the number of -1s. $\endgroup$ – Jack Schmidt May 6 '14 at 18:38
  • $\begingroup$ @ Jack Schmidt: I am trying to prove it. May I find the structures of 2-sylow subgroups of finite simple groups anywhere? $\endgroup$ – Adeleh May 7 '14 at 6:45
  • $\begingroup$ Yes, for classical groups: ams.org/mathscinet-getitem?mr=166271 -- I think something similar works for the other types, but I'm not sure if there is a uniform description for all the exceptional types. $\endgroup$ – Jack Schmidt May 7 '14 at 15:57
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Yes, though it is rare.

Lemma: Suppose $q=2^n \pm 1 > 3$ is a prime power. Then the simple group $\operatorname{PSL}(2,q)$ has a single class of involutions, and its centralizer is the Sylow 2-subgroup, a dihedral group of order $2^n$.

Example: PSL(2,7) = GL(3,2), PSL(2,9) = A6, PSL(2,17), PSL(2,31), PSL(2,127), PSL(2,257).

Lemma: The Suzuki groups have a single class of involutions, and its centralizer is the Sylow 2-subgroup (a specific type of Suzuki 2-group, necessarily non-abelian). Suzuki (1961)

Example: Sz(8), Sz(32)

Also there are a few weirdos:

Example: PSL(3,4) has a single class of involutions, and its centralizer is the Sylow 2-subgroup. PSp(4,4) has three classes of involutions (all central), and ONE of the classes has centralizer a Sylow 2-subgroup (the others contain the Sylow as index 15).

There are no other examples of order less than 300,000,000 of simple groups with non-abelian Sylow 2-subgroups and an involution whose centralizer is a 2-group. PSp(4,8) is similar to PSp(4,4). I haven't checked if it fits in an infinite family.

There are no examples in odd characteristic and high-rank: checking the table in this answer, the last column on page 172 and 174 includes a subgroup of the centralizer, and except for A1, it is not even a solvable group, much less a 2-group.

Suzuki (1961) classified the groups in which every involution's centralizer is a 2-group. The classification is particularly clear for non-abelian simple groups: $\operatorname{PSL}(2,p)$ for a $p$ a Fermat or Mersenne prime, $\operatorname{PSL}(2,9)$, $\operatorname{PSL}(3,4)$, $\operatorname{PSL}(2,2^n)$, or $\operatorname{Sz}(2^{2n+1})$. Note the $\operatorname{PSp}(4,2^n)$ don't appear here, since they have some good involutions and some bad ones.

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  • $\begingroup$ The discrepancy between primes and prime powers is explained in math.stackexchange.com/a/736708/583 $\endgroup$ – Jack Schmidt May 7 '14 at 18:00
  • $\begingroup$ @ jack Schmidt: Your answer is nice and useful. Thank you a lot. $\endgroup$ – Adeleh May 8 '14 at 5:32

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