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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
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