# Reference request for the number of Sylow p-subgroups

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.

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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.

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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.

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,

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".

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