Let $G$ be a finite group and let $X \subset G \setminus \{1\}$, with $n = |X|$. What I want to prove is the following:
Suppose that, for every prime $p < n$, the $p$-Sylow subgroups are cyclic. Then, there exists $\chi \in \operatorname{Irr}(G)$ such that $X \cap \ker\chi = \emptyset$.
I was struggling with the abelian case and took a look at the original paper ("A characterization of groups in terms of the degrees of their characters") and understood the proof. However, in the book, Isaacs gives the following hint for the abelian case:
Hint: Let $N$ be a maximal subgroup of $G$ with respect to the property $N \cap X = \emptyset$. Then show $G/N$ is cyclic.
Trying to prove this hint, I got nowhere. Contrary to the proof in the paper (which used induction), I have no control over the structure of the $N$ that appears here, since it isn't constructed from the ground-up.
My idea would be to show that every Sylow subgroup of the quotient $G/N$ is cyclic, but, since I don't even know the order of $N$, I don't know how to relate the Sylow subgroups in the quotient to those of the original group (they're the quotients of things of the form $PN$, but that's pretty much all I know).
Assuming $G = \langle X \rangle$ (WLOG), the quotient is generated by the projections of the $n$ elements $x_i \in X$, so another way to approach this would be to directly show that $x_1^\alpha x_j^{-\beta} \in N$ for some powers $\alpha, \beta$, but, again, I'm at a loss...
Do you have any hints as for how to prove the hint and get the result this way?
Thanks in advance!
PS: This isn't homework; I've taken it upon myself to solve all exercises in the first few chapters of the book and have no-one to ask for help...
Edit: clarified the request