Regular orbits of linear groups with an application to the \(k(GV)\)-problem. II
type: Item
instance of: scholarly article
title: Regular orbits of linear groups with an application to the \(k(GV)\)-problem. II
zbMATH Open document ID: 0970.20006
author: Dominic P. M. Goodwin
published in: Journal of Algebra
publication date: 2001-07-04T00:00:00Z
review text: This paper completes the research begun by the author in the first paper of this series, [J. Algebra 227, No. 2, 395-432 (2000; see the preceding review Zbl 0970.20005)]. Let \(p\) be a prime. Let \(G\) be a finite group of \(p'\)-order and let \(V\) be a faithful \(FG\)-module of finite dimension \(n\), where \(F\) is the field with \(p\) elements. Suppose that \(G\) contains a quasi-simple normal subgroup \(H\) that acts irreducibly on \(V\) and that \(H/Z(H)\) is a group of Lie type. Suppose also that \(G\) has no regular orbit on \(V\). Then \(H/Z(H)\), \(n\) and \(p\) belong to a finite list of possibilities, described in our earlier review. The author works his way systematically through the different families of groups of Lie type. The greatest problems in his analysis arise for symplectic and unitary groups, and tend to be associated with the case that \(V\) affords a Weil representation of \(H\). A certain number of rather specific module actions have to be examined in detail for a small number of groups. The author also makes good use of a paper of \textit{G. M. Seitz} and \textit{A. E. Zalesskij}, [J. Algebra 158, No. 1, 233-243 (1993; Zbl 0789.20014)], which gives lower bounds for the dimension of any non-trivial projective representation of a group of Lie type.
reviewed by: Rod Gow
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20C33
zbMATH DE Number: 1471027
zbMATH Keywords: groups of Lie type
zbMATH Keywords: regular orbits
zbMATH Keywords: Weil representations
zbMATH Keywords: faithful modules
zbMATH Keywords: irreducible group actions
zbMATH Keywords: \(p\)-solvable groups
MaRDI profile type: Publication
full work available at URL: jabr.1999.8198
OpenAlex ID: W2085296975
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DOI: 10.1006/JABR.1999.8198
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description: scientific article; zbMATH DE number 1471027