Exercise 22, § 6

Let $G$ be a finite group of order $n > 1$ and let $p$ be the least prime number dividing $n$. Let $P$ be a Sylow $p$-subgroup of $G$ and $N$ its normalizer. Show that, if $P$ is cyclic, $P$ is contained in the centre of $N$ (show that the order of $N/P$ is prime to the order of the automorphism group of $P$).