Finite groups admitting a \(p\)-solvable operator group
type: Item
instance of: scholarly article
title: Finite groups admitting a \(p\)-solvable operator group
zbMATH Open document ID: 0924.20012
published in: Journal of Algebra
publication date: 1999-11-09T00:00:00Z
review text: Consider a \(\pi\)-solvable operator group \(H\) acting on a group \(G\). The author proves that if \(G\) has a nilpotent maximal \(H\)-invariant subgroup which has a Sylow 2-subgroup with no section isomorphic to the dihedral group of order eight, then \(G\) is solvable. This generalizes results on nilpotent maximal subgroups by J. G. Thompson and Z. Janko. Also a group \(G\) is solvable if \(H\) acts nontrivially on \(G\) but trivially on each proper \(H\)-invariant subgroup, in fact \(G\) is metabelian with a well-defined structure. However if \(H\) acts trivially on each \(H\)-invariant metabelian subgroup of \(G\), then \(H\) acts trivially on \(G\). Several other similar results are developed and examples are provided to highlight the necessity for the constraints in the hypotheses.
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 1218686
zbMATH Keywords: \(\pi\)-solvable operator groups
zbMATH Keywords: nilpotent maximal \(H\)-invariant subgroups
zbMATH Keywords: nilpotent maximal subgroups
zbMATH Keywords: \(H\)-invariant metabelian subgroups
author: Yanming Wang
reviewed by: Homer F. Bechtell
MaRDI profile type: Publication
full work available at URL: jabr.1997.7351
OpenAlex ID: W2053721740
cites work: On finite simple groups of characteristic 2 type
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cites work: Verallgemeinerung eines Satzes von B. Huppert und J. G. Thompson
cites work: Endliche Gruppen I
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cites work: Character-theoretic transfer
DOI: 10.1006/JABR.1997.7351
description: scientific article; zbMATH DE number 1218686