Large orbits of subgroups of solvable linear groups.
type: Item
instance of: scholarly article
title: Large orbits of subgroups of solvable linear groups.
zbMATH Open document ID: 1302.20006
author: Yong Yang
published in: Israel Journal of Mathematics
publication date: 2014-06-25T00:00:00Z
review text: Let \(G\) be a finite group and \(V\) a finite, faithful and completely reducible \(G\)-module. It is a classical theme to study orbit structures of \(G \) acting on \(V\). One of the most important and natural questions about orbit structure is to establish the existence of an orbit of a certain size. In this paper, the author proves the following results. Suppose that \(G\) is a finite solvable group and \(V\) is a finite, faithful and completely reducible \(G \)-module. Let \(H\) be a nilpotent subgroup of \(G\); then \(H\) has at least three regular orbits on \(V\oplus V\). Let \(H\) be a subgroup of \(G\) and \(3\nmid |H|\); then \(H\) has at least three regular orbits on \(V\oplus V\). Let \( H\) be a subgroup of \(G\) and assume the Sylow \(2\)-subgroups of the semidirect product \(HV\) are Abelian; then \(H\) has at least two regular orbits on \(V\oplus V\).
reviewed by: Nan Ji-Zhu
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 6308384
zbMATH Keywords: finite solvable groups
zbMATH Keywords: completely reducible modules
zbMATH Keywords: regular orbits
zbMATH Keywords: numbers of orbits
describes a project that uses: GAP
MaRDI profile type: Publication
full work available at URL: s11856-014-0002-x
OpenAlex ID: W1991990247
cites work: INTERSECTIONS OF ODD ORDER HALL SUBGROUPS
cites work: Large orbits in coprime actions of solvable groups
cites work: Large character degrees of solvable groups with Abelian Sylow 2-subgroups.
cites work: Large character degrees of groups of odd order
cites work: The existence of regular orbits
cites work: Regular orbits of Hall \(\pi\)-subgroups
cites work: The largest Irreducible Character Degree of a Finite Group
cites work: Q4292200
cites work: Q4871631
cites work: Large orbits in actions of nilpotent groups
cites work: Q4273959
cites work: HEIGHTS OF CHARACTERS IN BLOCKS OF $p$-SOLVABLE GROUPS
cites work: Q4387775
cites work: Q4660848
cites work: Orbits of the actions of finite solvable groups.
cites work: Regular orbits of finite primitive solvable groups
cites work: Regular orbits of finite primitive solvable groups. II.
cites work: Large character degrees of solvable 3’-groups
cites work: Regular orbits of nilpotent subgroups of solvable linear groups.
DOI: 10.1007/S11856-014-0002-X
description: scientific article; zbMATH DE number 6308384