Prove Sylow-p Subgroups Abelian

Date: 05/15/2003 at 05:01:32
From: Noot
Subject: Modern algebra

Hi, Doctor Math. I have a puzzling question.

G is a finite simple group with exact 2p + 1 sylow-p subgroups. Prove 
that each of these sylow-p subgroups is Abelian.

Date: 05/16/2003 at 07:55:13
From: Doctor Jacques
Subject: Re: Modern algebra

Hi Noot,

The order of G is:

  |G| = p^k*(2p + 1)*m

where p does not divide m.

Any group of order p or p^2 (p prime) is Abelian, so we may assume 
that k >= 3.

Now, if we let G act by conjugation on the set of Sylow p-subgroups, 
we see that there is a homomorphism

  f : G -> S_(2p + 1)

As G is simple, this homomorphism is injective, and its image is a 
subgroup of order |G| in S_(2p + 1).

This should allow you to put some heavy restrictions on p, k, and m, 
and G itself.

Does this help?  Write back if you'd like to talk about this 
some more, or if you have any other questions.

- Doctor Jacques, The Math Forum
  http://mathforum.org/dr.math/