Reference request for the number of Sylow p-subgroups

Question

Let $G$ be simple group of Lie type or Alternating group. I need reference for find the number of Sylow $p$-subgroup $G$ for every $p$. Thanks a lot.

Answer 1

You could try

Mark Stather, Constructive Sylow theorems for the classical groups. J. Algebra 316 (2007), no. 2, 536–559.

This paper is basically about algorithms for constructing the Sylow $p$-subgroups of classical groups, but it includes descriptions of their normalizers, and also the normalizers of the Sylow subgroups of the symmetric groups, so you should be able to use that information to compute the orders of the normalizers, and hence the number of Sylow subgroups, but I am afraid there are lots of cases!

Of course, this builds on many earlier papers on this topic, such as

A.J. Weir, Sylow p-subgroups of the classical groups over finite fields with characteristic prime to p, Proc. Amer. Math. Soc. 6 (1955) 529–533.

R. Carter, P. Fong, The Sylow 2-subgroups of the finite classical groups, J. Algebra 1 (1964) 139–151.

Rimhak Ree, On some simple groups defined by C. Chevalley, Trans. Amer. Math. Soc. 84 (1957) 392–400.

I am afraid I cannot help with exceptional groups of Lie type.

Answer 2

I was just rushing over old mails from group-pub forum and saw a similar question there. I just copy the question and answer. It gives an alternate reference in the cross characteristic case to the ones given by Derek.

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Dear Forum, I was wondering if anyone knew of a place I could find orders of normalizers of Sylow subgroups of Lie groups? Best wishes,

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I presume you're referring to finite groups of Lie type - I have heard that "Sylow subgroup" has meaning outside of finite groups, but I'm guessing that it's finite groups that you mean.

Let L be a finite group of Lie type over a field of characteristic p. The normalizer of a Sylow p-subgroup is just a Borel subgroup; this is a standard fact to be found in any book on the subject.

For cross characteristic, i.e. for Sylow t-subgroups with t not equal to p, I'd refer you to Gorenstein, Lyons, Solomon, CSFG vol 3, Section 4.10 where this is discussed (for simple groups, but the generalization to other groups of Lie type isn't too hard I think).

Apart from small t (I think excluding t dividing the order of the Weyl group will be enough) the Sylow t-subgroup is a subgroup of a unique maximal torus of L and the normalizer of the Sylow t-subgroup is the normalizer of the maximal torus containing it.

Normalizers of maximal tori in groups of Lie type are also discussed at great length in Chapter 3 of Carter's "Finite Groups of Lie type".

Answer 3

It's probably useful to compare the other questions recently asked here by Sara, which are sometimes stimulating but typically not well formulated or motivated: for example here. Like the other questions, this one seems to be asking a very broad question which isn't easily answered for all simple groups. Here the number of Sylow subgroups is measured by the index of the normalizer, which itself requires a lot of case-by-case study. There certainly aren't any convenient tables of results to consult, though the various references given here and in other answers are a good start.

Leaving aside the alternating groups, there are two distinct questions for simple groups of Lie type: When the prime is the defining one, the internal structure of the group given by the BN-pair is a good guide to the normalizer of a Sylow subgroup (characterized in terms of algebraic groups via the unipotent radical of a Borel subgroup). But for other primes there are more subtle questions raised, which many people have studied in the setting of block theory and Deligne-Lusztig character theory. In the literature, it seems most natural to focus on the behavior of each prime according to which factor of the order polynomial for the group it divides. This work aims mostly at comparing the ordinary and the modular representation theory for the given prime, but I'm not sure how explicitly it involves the normalizer order or index wanted here. As Derek Holt points out, there may be extra detail available case-by-case in the case of finite classical groups.

Anyway, it's better to start with a more fully developed question which might get a precise answer.