On generalized Dedekind groups.
type: Item
instance of: scholarly article
title: On generalized Dedekind groups.
zbMATH Open document ID: 1188.20016
published in: Acta Mathematica Hungarica
publication date: 2009-12-29T00:00:00Z
review text: Let \(G\) be a finite group. The norm of the group \(G\) consists of the elements normalizing all subgroups, and has been the subject of research for a long time, and recently, for instance, by \textit{M. Herzog, P. Longobardi, M. Maj, A. Mann} [J. Algebra 226, No. 2, 690-713 (2000; Zbl 0959.20031)] and by the authors [Algebra Colloq. 14, No. 4, 605-612 (2007; Zbl 1149.20016)]. The group \(G\) is called an N-group if any of its elements either generates a normal subgroup or is in the norm. In the present paper the authors characterize N-groups, namely, these are exactly nilpotent groups with Sylow-subgroups being N-groups, and at most one of them is not a Dedekind (Hamiltonian) group. Moreover, N-groups of prime power order are also characterized, and as a consequence, one also obtains the characterization of groups in which any element of prime power order either generates a normal subgroup or is in the norm.
reviewed by: János Kurdics
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 5651928
zbMATH Keywords: generalized Dedekind groups
zbMATH Keywords: norm of finite groups
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: normalizers
author: Xiuyun Guo
author: Junxin Wang
MaRDI profile type: Publication
full work available at URL: s10474-009-7178-2
OpenAlex ID: W2023004068
cites work: Q4756764
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cites work: Centre and norm
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cites work: On the norm of a group
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DOI: 10.1007/S10474-009-7178-2
description: scientific article; zbMATH DE number 5651928