Element orders and codegrees of characters in non-solvable groups

Given a finite group $G$ and an irreducible complex character $\chi$ of $G$, the codegree of $\chi$ is defined as the integer $\mathrm{c}\mathrm{o}\mathrm{d}(\chi )=\mathrm{\mid }G:\ker \chi \mathrm{\mid }\mathrm{/}\chi (1)$. It was conjectured by G. Qian in [13] that, for every element $g$ of $G$, there exists an irreducible character $\chi$ of $G$ such that $\mathrm{c}\mathrm{o}\mathrm{d}(\chi )$ is a multiple of the order of $g$; the conjecture has been verified under the assumption that $G$ is solvable ([13]) or almost-simple ([11]). In this paper, we prove that Qian’s conjecture is true for every finite group whose Fitting subgroup is trivial, and we show that the analysis of the full conjecture can be reduced to groups having a solvable socle.