Regular orbits of finite primitive solvable groups. II.
type: Item
instance of: scholarly article
title: Regular orbits of finite primitive solvable groups. II.
zbMATH Open document ID: 1247.20021
author: Yong Yang
published in: Journal of Algebra
publication date: 2012-02-11T00:00:00Z
review text: Let \(G\) be a finite solvable group and suppose it acts faithfully, irreducibly and quasi-primitively on a finite vector space \(V\). Then \(G\) has a uniquely determined normal subgroup \(E\) which is a direct product of extraspecial \(p\)-groups for various primes \(p\), and we set \(e=\sqrt{|E/Z(E)|}\). In Part I of the paper [J. Algebra 323, No. 10, 2735-2755 (2010; Zbl 1195.20017)] the author proved that, if \(e\geq 10\) and \(e\neq 16\), then \(G\) has a regular orbit on \(V\). In the present paper, Yang proves that if \(e\in\{5,6,7\}\) then \(G\) also has a regular orbit on \(V\). He also shows that this together with the earlier result completes the list of all \(e\) for which there always is a regular orbit on \(V\). Some further information on the number of regular orbits is also obtained. In addition, some applications of these results are given. For Part III, see [the author et al., J. Algebra 590, 139-154 (2022; Zbl 07420015)].
reviewed by: Alexandre Turull
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20D25
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 6005424
zbMATH Keywords: representations of solvable groups
zbMATH Keywords: solvable linear groups
zbMATH Keywords: regular orbits
describes a project that uses: GAP
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2011.03.031
OpenAlex ID: W4213044664
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cites work: Regular orbits of finite primitive solvable groups
DOI: 10.1016/J.JALGEBRA.2011.03.031
description: scientific article; zbMATH DE number 6005424