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Paper IPM / M / 17413  


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 nontrivial normal subgroup of $G$.
Let $\acdk(G)$ and $\acdek(GN)$ be the average degree of all nonlinear $\Bbb F$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 $0<{\rm acd}_{\mathbb{Q},even}(GN)<4$, then $G$ is solvable.
Moreover, setting $\Bbb{F} \in \{\Bbb{R},\Bbb{C}\}$, we obtain the solvability of $G$ by assuming $\acdk(G)<29/8$ or $0<\acdek(GN)<7/2$, and we conclude the solvability of $N$ when $0<\acdek(GN)<18/5$. Replacing $N$ by $G$ in $\acdek(GN)$ gives us an extended form of a result by Moreto and Nguyen. Examples are given to show that all the bounds are sharp.
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