Let $G$ be a non-abelian finite simple group, let $p$ be a prime dividing the order of $G$, and let $P < G$ be a Sylow $p$-subgroup.

Is there a list of all the cases where $N_G(P)$ is maximal? If not, would it be feasible/difficult to list them all, using results from the literature on maximal subgroups of finite simple groups? Is there already a list somewhere in the literature?

If $G$ is sporadic group, I looked at the ATLAS and I think here are the cases where a Sylow $p$-normalizer is maximal in $G$ (hopefully I did not miss any cases):

  • M11, $p = 3$
  • M23, $p = 23$
  • J2, $p = 5$
  • Co1, $p = 7$
  • Co2, $p = 5$
  • McL, $p = 5$
  • He, $p = 5, 7$
  • Th, $p = 5, 7, 31$
  • Fi24', $p = 29$
  • B, $p = 47$
  • M, $p = 41$
  • J1, $p = 2, 3, 5, 7, 11, 19$ (every prime divisor of order)
  • O'N, $p = 3$
  • J3, $p = 3$
  • Ru, $p = 5$
  • J4, $p = 11, 29, 37, 43$
  • Ly, $p = 37, 67$
  • T, $p = 5$

I think the case where $G$ is an alternating group follows from the answer of Derek Holt here, using the classification of $2$-transitive groups.

  • $\operatorname{Alt}_5$, $p = 2,3,5$
  • $\operatorname{Alt}_6$, $p = 3$
  • $\operatorname{Alt}_p$, for $p = 13, 19$ or $p \geq 29$

So that leaves the simple groups of Lie type, of which there are many. The sporadic and alternating case, and computations in some small cases suggest that perhaps it is relatively rare for $N_G(P)$ to be maximal.

  • 3
    $\begingroup$ There is enough information about the maximal subgroups of the classical simple groups in the books by Kleidman and Liebeck for dimension greater than 12 and by Bray, Holt and Roney-Dougal for dimensions up to 12 to answer this question. As you suspect, it is unusual for a Sylow normalizer to be maximal, but it can happen. One set of examples are the normalizers of a Singer cycles in ${\rm PSL}(p,q) $ and ${\rm PSU}(p,q)$ for prime degrees $p$. That would leave the exceptional groups of Lie type and a lot is known about their maximal subgroups. $\endgroup$
    – Derek Holt
    Nov 10 '19 at 23:02
  • $\begingroup$ For groups of Lie type, it all depends on whether $p$ is the characteristic of the underlying field. If it is, then your normalizer is a Borel, and this will only be maximal in rank 1 groups ($PSL_2$, ${^2B_2}$, ${^2G_2}$). If $p$ is not the characteristic prime, then for the classicals it seems to me that you want to think about whether or not the Sylow is irreducible or not. If it isn't, then you stop -- the normalizer will be inside a parabolic, and can't be maximal. If it is irreducible, then it will (I think) lie inside a field-extension subgroup... $\endgroup$
    – Nick Gill
    Nov 11 '19 at 17:27
  • $\begingroup$ ... It can only be maximal if it lies in the centre of such a thing... We get the Singer examples that Derek mentioned, and I'd have to think some more if there are others. (By the way, Derek's method of using the results on maximals is perfectly valid -- I was just trying to work out if there was a more direct way.) $\endgroup$
    – Nick Gill
    Nov 11 '19 at 17:27
  • $\begingroup$ Incidentally, for $p=2$, an alternative approach would be to use the list of odd index maximals given in Liebeck/Saxl (and also in Kantor). You can quickly cross of any that aren't normalizers of Sylow $2$-subgroups. (Note that there is one missing family in those two papers -- it occurs in ${^2G_2}(q)$.) $\endgroup$
    – Nick Gill
    Nov 11 '19 at 17:29

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