Variations on average character degrees and solvability

Abstract

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