Variations on average character degrees and solvability

TL;DR

This paper investigates how bounds on average degrees of certain irreducible characters of a finite group can determine the group's solvability, extending previous results and providing sharp bounds with examples.

Contribution

It establishes new bounds on average character degrees that guarantee the solvability of finite groups and their normal subgroups, extending prior work by Moreto and Nguyen.

Findings

If ${ m acd}^*_{{Q}}(G)< 9/2$, then $G$ is solvable.

If $0<{ m acd}_{{Q},even}(G|N)<4$, then $G$ is solvable.

Bounds are sharp, with examples illustrating the limits.

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.