Exercise 22, § 6

Let $G$ be a finite group of order $n > 1$ and let $p$ be the smallest prime number dividing $n$. Let P be a Sylow $p$-group of G and let N be its normalizer. Now prove that, if P is cyclic, P is contained in the center of N (one will show that the order of N/P is prime to the order of the automorphism group of P).