Existence of a normal $p$-complement. (5C.12 Finite Group Theory, Isaacs)

Question

Let $G$ be a finite group, $N$ normal subgroup with index in $G$ divisible by $p$ prime and suppose that a Sylow $p$-subgroup of $G$ is cyclic. Then $N$ has a normal $p$-complement. This is the exercise 5C.12 of "Finite Group Theory", Isaacs. Can someone give any hint? Now I don't have any idea.

Answer 1

Let $P \in {\rm Syl}_p(G)$, $Q =N \cap P$. To apply Burnside's Transfer Theorem, you need to show that $Q$ is central in $N_N(Q)$. We may as well assume that $N = N_N(Q)$ and $G = PN$. Then $N=QH$ where $H$ is a $p$-complement of $Q$ in $N$.

Let $P = \langle g \rangle$. Then $H^g$ is another $p$-complement, so it is conjugate to $H$ by an element $h$ of $Q$. Then by replacing $g$ by $gh^{-1}$ (which also generates $P$), we get $H^g = H$. Let $p^k$ be the highest power of $p$ that divides $|G/N|$. Then $g^{p^k}$ generates $Q$and normalizes $H$ and hence $[H,Q] \le Q \cap H = 1$, and we are done.