## 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 $N_G(P)$.