On the normalizer problem for integral group rings of torsion groups
type: Item
instance of: scholarly article
title: On the normalizer problem for integral group rings of torsion groups
zbMATH Open document ID: 1016.20001
DOI: 10.1007/BF02784509
published in: Israel Journal of Mathematics
publication date: 2003-05-13T00:00:00Z
review text: Given a group \(G\), let \({\mathcal U}(\mathbb{Z} G)\) be the group of units of its integral group ring \(\mathbb{Z} G\). The normalizer \(N_{\mathcal U}(G)\) of \(G\) in \({\mathcal U}(\mathbb{Z} G)\) clearly contains \(G\) and the centre \({\mathcal Z}={\mathcal Z}({\mathcal U}(\mathbb{Z} G))\) of \({\mathcal U}(\mathbb{Z} G)\). The normalizer problem asks for a description of the quotient \(N_{\mathcal U}(G)/G{\mathcal Z}\). The group \(G\) is said to have the normalizer property if \(N_{\mathcal U}(G)=G{\mathcal Z}\), or equivalently if \(\Aut_{\mathcal U}(G)=\text{Inn}(G)\), where \(\Aut_{\mathcal U}(G)\) is the group of automorphisms of \(G\) induced by the elements of \({\mathcal U}(\mathbb{Z} G)\). Extending a result of \textit{D. B. Coleman} [Proc. Am. Math. Soc. 15, 511-514 (1964; Zbl 0132.27501)], the author proves that locally finite nilpotent groups have the normalizer property. The second main result proved in this paper is that a torsion group \(G\) has the normalizer property if for a maximal Sylow 2-subgroup \(P\) of \(G\), every involution \(\varphi\in\Aut_{\mathcal U}(G)\) which keeps \(P\) pointwise fixed is inner; this extends a result of \textit{S. Jackowski} and \textit{Z. Marciniak} [J. Pure Appl. Algebra 44, 241-250 (1987; Zbl 0624.20024)].
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20F28
Mathematics Subject Classification ID: 20F50
zbMATH DE Number: 1874391
zbMATH Keywords: normalizers
zbMATH Keywords: integral group rings
zbMATH Keywords: automorphism groups
zbMATH Keywords: torsion groups
zbMATH Keywords: groups of units
zbMATH Keywords: locally finite nilpotent groups
author: Yuanlin Li
reviewed by: Inder Bir S. Passi
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1874391