1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Congruence and p-sylow subgroups

  1. Nov 25, 2009 #1
    1. The problem statement, all variables and given/known data

    Let p denote a prime number. Let G be a finite group. Let P denote a p-sylow subgroup of G. Finally, let H be any subgroup of G that contains N_G(P). Prove that [G:H]≡1 mod p.

    2. Relevant equations

    Well I believe all we have to do here is conjugate stuff to show membership in various normalizers and then invoke Sylow III.

    3. The attempt at a solution

    First off we know that for g [tex]\in[/tex] NG(H) such that gPg-1 and gHg-1. We also know that there exists a p-sylow subgroup Q=gPg-1. Since Q is a p-sylow subgroup of H, there exists an h in H such that hPh-1=Q. Now we see that ghP(gh)-1=Q. Subbing in something new, j=gh, jPj-1 is in NH(P) which is in H. This also shows, by moving stuff around, that NG(H) is in H. So, now we conclude H is in the Normalizer of H and the Normalizer is in H, so H=NG(H). The by sylow III, we get that ratio. Is this right, or close?

    I was told there is a counting argument that uses group actions and this was more direct but a tad more complicated.
    Last edited: Nov 25, 2009
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted