On subnormal subgroups of factorized groups
type: Item
instance of: scholarly article
title: On subnormal subgroups of factorized groups
zbMATH Open document ID: 0888.20015
author: Silvana Franciosi
author: F. de Giovanni
author: Y. P. Sysak
published in: Journal of Algebra
publication date: 1998-03-24T00:00:00Z
review text: Let the group \(G=AB\) be the product of two subgroups \(A\) and \(B\), and let \(H\) be a subgroup of the intersection \(A\cap B\). A well-known result of Maier and Wielandt says that if \(G\) is finite and the subgroup \(H\) is subnormal in \(A\) and \(B\), then \(H\) is also subnormal in \(G\) [see the book ``Products of groups'', Oxford Univ. Press (1992; Zbl 0774.20001), by the first two authors and the reviewer]. Here the authors generalize this theorem to the case when the group \(G\) is nilpotent-by-abelian-by-finite, thus extending a result of Stonehewer for periodic groups of this type. The generalization includes in particular soluble-by-finite linear groups \(G\). The proofs in this paper involve some group ring methods of independent interest. In particular some facts about radical modules play an important role. Here a \(G\)-module \(M\) is radical if and only if the augmentation ideal \(\Delta_\mathbb{Z}(G)\) of the integral group ring \(\mathbb{Z} G\) contains a one-sided ideal \(P\) such that \(\Delta_\mathbb{Z}(G)=G-1+P\) and \(M\) is isomorphic to the \(G\)-module \(\Delta_\mathbb{Z}(G)/P\).
Mathematics Subject Classification ID: 20E15
Mathematics Subject Classification ID: 20E22
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 16S34
zbMATH DE Number: 1106620
zbMATH Keywords: subnormal subgroups
zbMATH Keywords: products of subgroups
zbMATH Keywords: factorized groups
zbMATH Keywords: soluble groups
zbMATH Keywords: nilpotent-by-Abelian-by-finite groups
zbMATH Keywords: soluble-by-finite linear groups
zbMATH Keywords: integral group rings
zbMATH Keywords: radical modules
zbMATH Keywords: augmentation ideals
reviewed by: Bernhard Amberg
MaRDI profile type: Publication
full work available at URL: jabr.1997.7150
OpenAlex ID: W2048239833
cites work: Q3134546
cites work: On ascending and subnormal subgroups of infinite factorized groups
cites work: Q3744347
cites work: Um problema da teoria dos subgrupos subnormais
cites work: Observations on group rings
cites work: Q3961021
cites work: Q3029188
cites work: Q4333343
cites work: Subnormalität in faktorisierten endlichen Gruppen
cites work: Soluble products of minimax groups, and nearly surjective derivations
DOI: 10.1006/JABR.1997.7150
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description: scientific article; zbMATH DE number 1106620