- #1

tom.young84

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## Homework Statement

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]≡1 mod p.

## Homework Equations

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

## The Attempt at a Solution

First off we know that for g [tex]\in[/tex] N

_{G}(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 N

_{H}(P) which is in H. This also shows, by moving stuff around, that N

_{G}(H) is in H. So, now we conclude H is in the Normalizer of H and the Normalizer is in H, so H=N

_{G}(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.

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