On \(p\)-parts of monomial character degrees of solvable groups (Q6545059)

Let \(G\) be a finite group, \(\mathrm{Irr}(G)\) the set of irreducible complex characters of \(G\) and let \(e_{p}(G)\) be the largest integer such that \(p^{e_{p}(G)}\) divides \(\chi(1)\) for some \(\chi \in \mathrm{Irr}(G)\). \textit{M. L. Lewis} et al. [J. Algebra 411, 182--190 (2014; Zbl 1305.20007)] showed that if \(G\) is solvable and \(e_{p}(G) \leq 1\) then \(|G: F(G)| \leq p^{2}\).\N\NAn element of \(\mathrm{Irr}(G)\) is monomial if it is induced from a linear character of a subgroup of \(G\).\N\NThe main result of paper under review is Theorem 1.2: Let \(G\) be a solvable group and \(p\) be a prime. Assume that \(p^{2} \nmid \chi(1)\) for every irreducible monomial character \(\chi\) of \(G\). (1) If \(p = 2\) or \(p \geq 5\) then \(|G:F(G)|_{p} \leq p^{2}\). (2) If \(p=3\) and \(O^{3'}(G)=G\) then \(|G : F(G)|_{3} \leq 9\). This result provides an affirmative answer to a question posed in [\textit{D. Rossi}, Arch. Math. 120, No. 4, 339--347 (2023; Zbl 1516.20034)].