Exercise 24, § 6

Let $G$ be a finite group, $H$ a Sylow subgroup of $G$ and $N$ the normalizer of $H$. Let $X_1, X_2$ be two subsets of the centre of $H$ and $s \in G$ such that $sX_1s^{-1} = X_2$. Show that there exists $n \in N$ such that $nxn^{-1} = sx_1s^{-1}$ for all $x \in X_1$ (apply the theorem on the conjugacy of Sylow subgroups to the centralizer of $X_1$). Deduce that two central elements of $H$ are conjugate in $G$ if and only if they are so in $N$.