Regular orbits of finite primitive solvable groups
type: Item
instance of: scholarly article
title: Regular orbits of finite primitive solvable groups
zbMATH Open document ID: 1195.20017
author: Yong Yang
published in: Journal of Algebra
publication date: 2010-06-08T00:00:00Z
review text: Let \(G\) be a finite primitive solvable irreducible linear group over some finite field. Let \(V\) be the finite vector space on which \(G\) acts naturally. An element \(v\in V\) generates a `regular orbit' if the only element of \(G\) which fixes \(v\) is the identity. The paper proves that unless \(G\) is very uncomplicated there will always be regular orbits of \(G\). More precisely, let us denote by \(F(G)\) the Fitting subgroup of \(G\), and by \(E=[F(G),G,\dots,G]\) the commutator subgroup of \(F(G)\) with \(G\) and the result again with \(G\), with the process repeated until the obtained subgroup ceases to decrease. Then, we denote by \(e=\sqrt{|E/Z(E)|}\), the square root of the order of the quotient of \(E\) by its center. It turns out that \(e\) is always a positive integer, and larger integers roughly correspond to more complicated groups. The following is proved in the paper. If \(e\geq 10\) and \(e\neq 16\), then \(G\) has a regular orbit. Furthermore, it is proved that if \(e\in\{8,9,16\}\), then \(G\) need not have a regular orbit. This strengthens a result of \textit{O. Manz} and \textit{T. R. Wolf} [Representations of solvable groups. Cambridge: Cambridge University Press (1993; Zbl 0928.20008)], where the existence of regular orbits is proved when \(e>118\). For Part II, see [the author, J. Algebra 341, No. 1, 23--34 (2011; Zbl 1247.20021)].
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: 5718178
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.2010.02.031
OpenAlex ID: W4213153669
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DOI: 10.1016/J.JALGEBRA.2010.02.031
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description: scientific article; zbMATH DE number 5718178