Minimal non-\(\mathcal F\)-groups.
type: Item
instance of: scholarly article
title: Minimal non-\(\mathcal F\)-groups.
zbMATH Open document ID: 1214.20019
published in: Ricerche di Matematica
publication date: 2009-12-14T00:00:00Z
review text: Let \(\mathcal F\) be a subgroup closed saturated formation of groups. \textit{V. N. Semenchuk} [Algebra Logika 18, 348-382 (1979; Zbl 0463.20018); translation in Algebra Logic 18, 214-233 (1980)] has described the minimal non-\(\mathcal F\)-groups. The present paper describes in great detail the minimal non-\(\mathcal F\)-groups, where \(\mathcal F\) is the class of extensions of Abelian groups by groups from a saturated subgroup closed formation (Theorem 2). \textit{I. D. Macdonald} [Math. Z. 76, 270-282 (1961; Zbl 0104.02202)] has characterised those groups whose three-generator subgroups are metabelian. The authors obtain the following result: If \(G\) is a finite soluble non-metabelian group all of whose proper subgroups are metabelian, then either \(G\) can be generated by 3 elements or \(G\) is a 2-group (Theorem 3).
reviewed by: Hans Lausch
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 5646586
zbMATH Keywords: saturated formations
zbMATH Keywords: minimal non-\(\mathcal F\)-groups
zbMATH Keywords: minimal non-metabelian groups
author: J. C. Beidleman
author: Hermann Heineken
MaRDI profile type: Publication
full work available at URL: s11587-009-0044-2
OpenAlex ID: W1977175622
cites work: Extreme classes of finite soluble groups
cites work: Minimal nicht überauflösbare, endliche Gruppen
cites work: Finite soluble groups
cites work: On certain varieties of groups
cites work: Q3913578
DOI: 10.1007/S11587-009-0044-2
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description: scientific article; zbMATH DE number 5646586