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Paper   IPM / M / 17618
School of Mathematics
  Title:   Element orders and codegrees of characters in non-solvable groups
  Author(s):  Zeinab Akhlaghi (Joint with E. Pacifici and L. Sanus)
  Status:   Published
  Journal: J. Algebra
  Vol.:  244
  Year:  2024
  Pages:   DOI: 10.1016/j.jalgebra.2024.01.011
  Supported by:  IPM
Given a finite group G and an irreducible complex character χ of G, the codegree of χ 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 χ of G such that 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.

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