Orbit sizes, character degrees and Sylow subgroups. (Q1826873)

Let \(G\) be a finite group. The \(i\)-th Fitting subgroup is inductively defined by \(F_{i+1}(G)/F_i(G)=\) Fitting subgroup of \(G/F_i(G)\), where \(F_1(G)\) stands for the Fitting subgroup of \(G\). The authors show some significant connections between lengths of conjugacy classes of \(G\), degrees of irreducible characters of \(G\), and the ascending Fitting series \(G\geq\cdots\geq F_i(G)\geq\cdots\geq F_1(G)\). We give an overview of their results, without being complete in respect to all the results obtained in the paper. Part of Theorem A (paraphrased): If \(|F_{10}(G)|\) is odd, then there exists a product of three distinct irreducible characters \(\chi_1\), \(\chi_2\), \(\chi_3\) of \(G\) such that \(\chi_1\chi_2\chi_3(1)\) is a multiple of \(|G/F_1(G)|\). Theorem C: If \(G\) is solvable, then there exists \(\mu\in\text{Irr}(F_{10}(G))\) such that \(\mu^G=\chi\in\text{Irr}(G)\). So \(|G:F_{10}(G)|\) divides \(\chi(1)\). Theorem E: If \(V\) is a finite completely reducible faithful \(G\)-module for a solvable group \(G\), then (1) there exists \(v\in V\) such that \(C_G(v)\leq F_9(G)\), (2) if \(|GV|\) is odd, there exists \(v\in V\) in a regular orbit of \(F_1(G)\) such that \(C_G(v)\leq F_2(G)\). As applications the authors provide results on zeroes of characters, the height-zero-conjecture, Huppert's \(\rho\)-\(\sigma\) conjecture, large Abelian subgroups and large orbit sizes, large orbits in actions of nilpotent groups. All together, a wealth of nice new results is presented.