Normal subsets of a Sylow subgroup which are conjugate in the supergroup are conjugate in the Sylow normalizer

Let A and B be normal subsets of a Sylow p-subgroup P \leq G. Prove that if A and B are conjugate in G then they are conjugate in N_G(P).


[With generous hints from Wikipedia.]

Suppose B = gAg^{-1}, with g \in G. Now since P \leq N_G(A), gPg^{-1} \leq N_G(gAg^{-1}) = N_G(B).

Note that P, gPg^{-1} \leq N_G(B) are Sylow, and thus are conjugate in N_G(B). So for some h \in N_G(B), we have P = hgPg^{-1}h^{-1}, so that hg \in N_G(P). Moreover, hgAg^{-1}h^{-1} = hBh^{-1} = B. Thus A and B are conjugate in N_G(P).

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