Exercise 24, § 6

Let $G$ be a finite group, let $H$ be a Sylow subgroup of $G$ and let $N$ be the normalizer of $H$. Let $X_1, X_2$ be two subsets of the center of $H$, and let $s \in G$ such that $sX_1s^{-1} = X_2$. Now prove that there exists $n \in \mathbf{N}$ such that $nxn^{-1} = sx(s^{-1})$ for all $x \in X_1$ (apply the theorem of 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 $N$.