Finite groups with trivial Frattini subgroup.
type: Item
instance of: scholarly article
title: Finite groups with trivial Frattini subgroup.
zbMATH Open document ID: 1033.20017
author: Luise-Charlotte Kappe
author: Joseph Kirtland
published in: Archiv der Mathematik
publication date: 2003-08-18T00:00:00Z
review text: The authors continue their investigations on \(xP\)-groups, that is groups in which each nontrivial \(x\)-subgroup has property \(P\), especially whenever \(x=a,n\), or \(c\) for arbitrary, normal, or characteristic subgroup respectively and \(P=C,S,PNS\), or \(CS\) for the respective types that each \(x\)-subgroup has a complement, proper supplement, a proper normal supplement, or a proper characteristic supplement. The investigation here focuses on \(nS\)-groups which of course includes the \(nC\)-groups about which much is known. Many analogous results are obtained due to the involvement of the Frattini subgroup. Among these results are characterizations of the supersoluble \(nS\)-groups, those soluble groups in which each proper epimorphic image is an \(nS\)-group, and \(nS\)-groups which contain an Abelian normal subgroup that has an Abelian quotient with trivial Frattini subgroup. The article concludes with an examination of conditions for normal subgroup supplementation, for example if a subgroup \(N\) normal in a group \(G\) is contained in a normal Sylow \(p\)-subgroup \(B\) of \(G\) such that \(N\) has a supplement in \(B\), then \(N\) has a supplement in \(G\). This paper supplements an earlier work of the authors [Glasg. Math. J. 42, No. 1, 37-50 (2000; Zbl 0945.20016)].
Mathematics Subject Classification ID: 20D25
Mathematics Subject Classification ID: 20D40
zbMATH DE Number: 1966443
zbMATH Keywords: supplemented subgroups
zbMATH Keywords: characteristic subgroups
zbMATH Keywords: normal complements
zbMATH Keywords: Frattini subgroup
zbMATH Keywords: supersoluble groups
zbMATH Keywords: Sylow subgroups
reviewed by: Homer F. Bechtell
MaRDI profile type: Publication
full work available at URL: s00013-003-0788-y
OpenAlex ID: W84814388
DOI: 10.1007/S00013-003-0788-Y
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description: scientific article; zbMATH DE number 1966443