A characterization of finite groups in terms of the codegrees of their characters (Q2150902)

Let \(G\) be a finite group and let \(\chi \in \mathrm{Irr}(G)\), the codegree of \(\chi\) is the number \(\mathrm{cod}(\chi)=|G: \ker \chi|/\chi(1)\). In [Commun. Algebra 27, No. 3, 1053--1056 (1999; Zbl 0929.20010)], \textit{S. M. Gagola jun.} and \textit{M. L. Lewis} showed that \(G\) is nilpotent if and only if \(\mathrm{cod}(\chi) \mid \chi(1)\) for every \(\chi \in \mathrm{Irr}(G)\). The paper under review is devoted to classifying the groups having exactly one irreducible character \(\chi\) such that \(\mathrm{cod}(\chi) \nmid \chi(1)\) (a detailed list is given in Theorem A whose statement is too long to be reported here).