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 ${\rm acd}<em>{\Bbb{F}}<sup>{*}(G)$ and ${\rm acd}</em>{\Bbb{F},even}(G|N)$ 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 ${\rm acd}^{*_{\mathbb{Q}}(G)&lt;} 9/2$ or $0&lt;{\rm acd}<em>{\mathbb{Q},even}(G|N)&lt;4$, then $G$ is solvable. Moreover, setting $\Bbb{F} \in {\Bbb{R},\Bbb{C}}$, we obtain the solvability of $G$ by assuming ${\rm acd}</em>{\Bbb{F}}<sup>{*}(G)&lt;29/8$ or $0&lt;{\rm acd}<em>{\Bbb{F},even}(G|N)&lt;7/2$, and we conclude the solvability of $N$ when $0&lt;{\rm acd}</em>{\Bbb{F},even}(G|N)&lt;18/5$. 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.