Solvability and supersolvability criteria related to character codegrees (Q6607142)

Let \(G\) be a finite group and let \(\mathrm{Irr}(G)\) be the set of irreducible characters of \(G\). If \(\chi \in \mathrm{Irr}(G)\), then the codegree of \(\chi\) is defined by \(\mathrm{cod}(\chi)=|G: \ker \chi|/\chi(1)\). \textit{S. M. Gagola} and \textit{M. L. Lewis} [Commun. Algebra 27, No. 3, 1053--1056 (1999; Zbl 0929.20010)] proved that \(G\) is nilpotent if and only if \(\chi(1)\) divides \(\mathrm{cod}(\chi)\) for all \(\chi \in \mathrm{Irr}(G)\).\N\NIt is known that a finite non-abelian simple group does not have a character codegree which is a prime power [\textit{D. Chillag} and \textit{M. Herzog}, Arch. Math. 55, No. 1, 25--29 (1990; Zbl 0671.20007)]. Moreover, \textit{F. Alizadeh} et al. [Commun. Algebra 49, No. 2, 538--544 (2021; Zbl 1498.20016)] proved that if \(G\) be a finite group with exactly one composite character codegree, then \(G\) is solvable.\N\NIn this vein, the following results are proved:\N\NTheorem A: Let \(G\) be a finite group with at most two composite character codegrees. Then \(G\) is solvable.\N\NTheorem B: Let \(G\) be a finite group. If \(\mathrm{cod}(\chi) \leq p_{\chi}(\chi(1)+1)\) for all non-linear \(\chi \in \mathrm{Irr}(G)\), where \(p_{\chi}\) is the largest prime divisor of \(|G: \ker \chi|\), then \(G\) is solvable.