Variations on average character degrees and solvability

Abstract

Let G be a finite group, \(\mathbb {F}\) be one of the fields \(\mathbb {Q},\mathbb {R}\) or \(\mathbb {C}\), and N be a non-trivial normal subgroup of G. Let \({\textrm{acd}}^{*}_{{\mathbb {F}}}(G)\) and \({\textrm{acd}}_{{\mathbb {F}}, \textrm{even}}(G|N)\) be the average degree of all non-linear \(\mathbb {F}\)-valued irreducible characters of G and of even degree \(\mathbb {F}\)-valued irreducible characters of G whose kernels do not contain N, respectively. We assume the average of an empty set is zero for more convenience. In this paper we prove that if \(\textrm{acd}^*_{\mathbb {Q}}(G)< 9/2\) or \(0<\textrm{acd}_{\mathbb {Q},\textrm{even}}(G|N)<4\), then G is solvable. Moreover, setting \(\mathbb {F} \in \{\mathbb {R},\mathbb {C}\}\), we obtain the solvability of G by assuming \({\textrm{acd}}^{*}_{{\mathbb {F}}}(G)<29/8\) or \(0<{\textrm{acd}}_{{\mathbb {F}}, \textrm{even}}(G|N)<7/2\), and we conclude the solvability of N when \(0<{\textrm{acd}}_{{\mathbb {F}}, \textrm{even}}(G|N)<18/5\). Replacing N by G in \({\textrm{acd}}_{{\mathbb {F}}, \textrm{even}}(G|N)\) gives us an extended form of a result by Moreto and Nguyen. Examples are given to show that all the bounds are sharp.