- #1

mathgirl313

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- 0

## Homework Statement

Now, G is a finite group, and P is a Sylow P-Subgroup of G, and H is a subgroup of G with P≤H≤G. So if P is normal in H, and H is normal in G, then P is normal in G.

## Homework Equations

## The Attempt at a Solution

I know I want to show that gpg^-1 is contained in P but that's about as far as I can get. I know gPg^-1 is going to be another Sylow P-Subgroup, but other than that I'm kind of stuck. When I try going about showing that gpg^-1 q, where q is an element in p, I have trouble showing the two elements are in P. I get as far as q=hph^-1 since q is in P and i think my best bet for showing it is contained in P is for closure...but that's all I get. I know I need to use the P is Sylow P-Subgroup since the property normally isn't transitive, but I'm not sure how to.

Thanks for any help.