Prove Sylow-p Subgroups AbelianDate: 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/ |
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