20.65 (2022)

Given a complex irreducible character $\chi \in \text{Irr}(G)$ of a finite group $G$, let $\mathbb{Q}(\chi)$ denote the field extension of $\mathbb{Q}$ obtained by adjoining to $\mathbb{Q}$ all the values of $\chi$. We say that a finite group $G$ is $k$-rational if $|\mathbb{Q}(\chi) : \mathbb{Q}|$ divides $k$ for every $\chi \in \text{Irr}(G)$. Does there exist a real-valued function $f$ such that if $p$ is the order of a cyclic composition factor of a $k$-rational group $G$, then $p \leqslant f(k)$?

If $k = 1$, then we know that $p \leqslant 11$ by a theorem of J. G. Thompson (J. Algebra, 319 (2008), 558–594).