# How to find more (finite almost simple) groups with a given Sylow subgroup

I'm looking for some examples of actions on Sylow p-groups, and often those actions appear in the case of finite almost simple groups. Given a finite almost simple group, I understand in principle how to calculate its Sylow p-subgroups (here p is usually 2 or 3), but perhaps I am just too slow at doing it. In particular, I am familiar with the papers of Weir and Carter-Fong.

I am not sure how to do the reverse calculation: given a p-group P (possibly described as "the Sylow p-subgroup of the almost simple group X", and X is something explicit like "PGL(3,19)" or "M11"), find all of the almost simple groups that have a Sylow p-subgroup isomorphic to P.

I am pretty sure some people know how to do this, but it's not really clear to me how to go about it. For instance, it would have never occurred to me that PSU(3,8), PSL(3,19), and 3D4(2) have isomorphic Sylow 3-subgroups.

Is there a description of how this is done?

I think there is likely to be a finite answer to the question: Obviously only finitely many (and probably O(p)) alternating groups could work for a given P. We take for granted that only finitely many sporadic groups could work. It seems that, similarly to the alternating case, there are only finitely many ranks (again probably O(p)) of Lie groups that could work, and hopefully for each Lie type (and rank), there are just some congruences on "q" that indicate which ones work and which don't.

However, I've not had much luck doing this calculation in examples, and so I am looking for papers or textbooks where this has been done. I have found some that state the result of doing something like this (post CFSG), and I have found several that do this in quite some detail, but pre-CFSG so they spend hundreds of pages eliminating impossible groups obscuring what should now be an easy calculation. I'm looking for something with the pedagogical style of the pre-CFSG papers, but that doesn't mind using the standard 21st century tools.

Alternatively: does anyone know of a vaguely feasible approach to construct all groups with given Sylow subgroups? Blackburn et al.'s Enumerating book has some upper bounds, but they are pretty outrageous and don't seem adaptable to a feasible algorithm for my problem.

If you are only interested in actions on the Sylow p-subgroup, you may not need to find all groups, but just all so-called p-fusion patterns, as formalized in the theory of fusion systems due to Puig, and developed further by Broto-Levi-Oliver and others.

See Broto, Carles; Levi, Ran; Oliver, Bob The homotopy theory of fusion systems. J. Amer. Math. Soc. 16 (2003), no. 4, 779--856, or survey articles eg by Markus Linckelmann or Broto-Levi-Oliver.

Finding the p-fusion patterns on a given p-group is also, implicitly or explicitly, a step in finding all (finite almost simple) groups with a given Sylow p-subgroup. (There is not a one-size-fits-all method for doing this, however...)

For a list of equivalences of different p-fusion patterns in the finite groups of Lie type see:

http://www.maths.abdn.ac.uk/~bensondj/html/archive/broto-moeller-oliver.html

which contains some of the examples you mention.

The proof uses homotopy theory, and there does not appear to be a purely group theoretic proof yet (!)

• Perfect. That was actually the point. I've been splitting the fusion systems by hand, but had no systematic method. This paper looks perfect. – Jack Schmidt Jun 5 '10 at 19:29

Number 3 (1998) in the series of AMS monographs by Gorenstein, Lyons, Solomon The Classification of the Finite Simple Groups is a useful general reference for the Sylow subgroups and related structure in finite groups of Lie type (see for example 3.3 and 4.10). The study of "local" subgroups including normalizers of Sylow subgroups plays a large role in finite group theory (especially CFSG) and related character theory over fields of positive characteristic inspired by Brauer, Alperin, Broue, and others. Which means there is a lot of literature, usually not directed toward your specific questions but maybe relevant.

Especially for small primes it's realistic to look for concrete information, but on the other hand the question raised in your last paragraph may be too broad even for small primes. (The word all is risky in this subject.) And there is a dwindling band of real specialists to consult about the structure of finite groups after the apparent success of CFSG.

There is a sense in which, for primes p >3, "most" finite p-groups can NOT occur as Sylow p-subgroups of finite simple groups (I know you asked about almost simple groups). For example, for such primes p, George Glauberman proved that a finite p-group P whose outer automorphism group is a p-group can not be the Sylow p-subgroup of a finite simple group. There are various theorems which show that "most" finite p-groups have no outer automorphism of order prime to p. But I agree with previous comments. Your last question sounds very very general.