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 cod(chi) = |G : ker(chi)|/chi(1). It was conjectured by G. Qian in [16] that, for every element g of G, there exists an irreducible character chi of G such that cod(chi) is a multiple of the order of g; the conjecture has been verified under the assumption that G is solvable ([16]) or almost-simple ([13]). 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. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY -NC -ND license (http://creativecommons .org /licenses /by -nc -nd /4 .0/).