Infinite groups with short balanced chains of subgroups.
type: Item
instance of: scholarly article
title: Infinite groups with short balanced chains of subgroups.
zbMATH Open document ID: 1151.20022
author: L. A. Kurdachenko
author: Alessio Russo
author: Igor Subbotin
author: Giovanni Vincenzi
published in: Journal of Algebra
publication date: 2008-07-21T00:00:00Z
review text: A subgroup \(H\) of a group \(G\) is abnormal in \(G\) if \(g\in\langle H,H^g\rangle\) for each element \(g\in G\). Abnormal subgroups are of interest because if \(H\leq K\leq G\) with \(H\) abnormal in \(G\) then \(K\) is self-normalizing. In this paper the authors obtain a number of interesting results connected to abnormality. A group \(G\) is a \(T\)-group if normality is a transitive relation and \(G\) is a \(\overline T\)-group if every subgroup of \(G\) is a \(T\)-group. It is well-known that \(\overline T\lneqq T\). In part of this paper the authors obtain new characterizations of \(\overline T\)-groups. For example they prove that if \(G\) is a radical group then \(G\) is a \(\overline T\)-group if and only if every cyclic subgroup of \(G\) is abnormal in its normal closure. Along the way to proving this result the authors establish numerous additional facts concerning groups all of whose cyclic subgroups are abnormal in their normal closures. For example, \textit{J. S. Rose} [Math. Z. 90, 29-40 (1965; Zbl 0131.02403)] defined a balanced chain connecting a subgroup \(H\) to a group \(G\) as a chain of subgroups \(H=H_0\leq H_1\leq\cdots\leq H_{n-1}\leq H_n=G\) such that, for each \(j\), either \(H_j\triangleleft H_{j+1}\) or \(H_j\) is abnormal in \(H_{j+1}\). Thus the groups all of whose cyclic subgroups are abnormal in their normal closures are examples of groups in which the cyclic subgroups have a balanced chain of length \(2\). Groups \(G\) all of whose subgroups are normal or abnormal (i.e. have balanced chains of length at most \(1\)) have been studied by \textit{A. Fattahi} [J. Algebra 28, 15-19 (1974; Zbl 0274.20022)], \textit{I. Ya. Subbotin} [Izv. Vyssh. Uchebn. Zaved., Mat. 1992, No. 3(358), 86-88 (1992; Zbl 0816.20030)] and \textit{M. De Falco, L. A. Kurdachenko} and \textit{I. Ya. Subbotin} [Atti Semin. Mat. Fis. Univ. Modena 46, No. 2, 435-442 (1998; Zbl 0918.20017)] and the work under review represents the next step in a classification theor…
Mathematics Subject Classification ID: 20E15
Mathematics Subject Classification ID: 20F19
zbMATH DE Number: 5303138
zbMATH Keywords: abnormal subgroups
zbMATH Keywords: pronormal subgroups
zbMATH Keywords: soluble groups
zbMATH Keywords: balanced chains
zbMATH Keywords: normal closures
zbMATH Keywords: transitive normality
reviewed by: Martyn Dixon
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2007.12.027
OpenAlex ID: W2031631882
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DOI: 10.1016/J.JALGEBRA.2007.12.027
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description: scientific article; zbMATH DE number 5303138