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Paper IPM / M / 17413

School of Mathematics
Title:	Variations on average character degrees and solvability
Author(s):	Zeinab Akhlaghi (Joint with N. Ahanjideh and K. Aziziheris)
Status:	To Appear
Journal:	Annali di Matematica Pura ed Applicata
Supported by:	IPM

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

$\acdk(G)$ and

$\acdek(G|N)$ 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

$\acdk(G)<29/8$ or

$0<\acdek(G|N)<7/2$ , and we conclude the solvability of

$N$ when

$0<\acdek(G|N)<18/5$ . Replacing

$N$ by

$G$ in

$\acdek(G|N)$ gives us an extended form of a result by Moreto and Nguyen. Examples are given to show that all the bounds are sharp.

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“School of Mathematics”

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