Proving that N(N(P)) = N(P) for p-Sylow Subgroups of G

[basukinjal]

If H is a subgroup of G then N(H) is defined as { x belonging to G | xHx^-1 = H }. If P is p-Sylow subgroup of G, then prove that N(N(P)) = N(P).

 

[donk]

If H is a subgroup of G then N(H) is defined as { x belonging to G | xHx^-1 = H }. If P is p-Sylow subgroup of G, then prove that N(N(P)) = N(P).

You need to prove both directions.
N(P) < N(N(P)) is obvious.

To show N(N(P)) < N(P), you might need to use the fact that P is normal in N(P), which implies the only Sylow p-subgroup of N(P) is P itself.