The normalizer of the normalizer of a p-sylow supgroup

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Im trying to prove N(N(P)) = N(P)

So N(P) = set oh h where h^-1Ph = p

Then N(N(P)) = k where k^-1hk = h

the fact that p is a p sylow subgroup gives me what information? I am unsure.

Thanks in advance!
 
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Well I guess if you can prove that one is a subset of the other, and both of them have the same cardinality, then they are equal.

Do you see how you can prove this?
 
Not really sorry. Both are subgroups of G yes?

If I let H = the normalizer of P
Can I say P is normal in H and that as all sylow subgroups are conjugate H contains no other sylow subgroups then if N(N(P)) moves P somewhere else then there would be 2 P sylow subgroups in H?

Note: Havent been told that P is normal in HThanks very much for you reply
 
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If P is the only subgroup of H, then P is normal in H
 
Thanks! So I can do it this way then with that result.
 

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