A generalization of Schmidt's theorem on groups with all subgroups nilpotent.
type: Item
instance of: scholarly article
title: A generalization of Schmidt's theorem on groups with all subgroups nilpotent.
zbMATH Open document ID: 1272.20023
published in: Archiv der Mathematik
publication date: 2012-11-21T00:00:00Z
review text: The author studies finite \(\mathcal S^n\)-groups, which are defined as the groups having exactly \(n\) non-nilpotent subgroups. This generalises the minimal non-nilpotent groups or Schmidt groups, which turn out to be the \(\mathcal S^1\)-groups. The main results of the paper are that finite \(\mathcal S^n\)-groups with \(n<22\) are soluble and that the alternating group of degree~\(5\) is an \(\mathcal S^{22}\)-group, and that the only finite non-Abelian semisimple (that is, with no non-trivial normal Abelian subgroups) \(\mathcal S^n\)-groups with \(n\leq 65\) are the alternating and the symmetric group of degree~\(5\) and the simple group of order~\(168\).
reviewed by: R. Esteban-Romero
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D05
zbMATH DE Number: 6109275
zbMATH Keywords: finite groups
zbMATH Keywords: Shmidt groups
zbMATH Keywords: numbers of non-nilpotent subgroups
zbMATH Keywords: minimal non-nilpotent groups
zbMATH Keywords: solvable groups
zbMATH Keywords: solvability
zbMATH Keywords: finite simple groups
author: Mohammad Zarrin
describes a project that uses: GAP
MaRDI profile type: Publication
full work available at URL: s00013-012-0411-1
OpenAlex ID: W2060206201
cites work: On finite minimal non-nilpotent groups
cites work: Semisimple groups with the rewriting property Q5
cites work: Insoluble groups with the rewriting property \(P_ 8\)
cites work: Q3251484
cites work: Groups with many nilpotent subgroups
cites work: Q3240869
cites work: Q4858090
cites work: Nonsolvable finite groups all of whose local subgroups are solvable
DOI: 10.1007/S00013-012-0411-1
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description: scientific article; zbMATH DE number 6109275