Multiplicative Jordan decomposition in group rings and \(p\)-groups with all noncyclic subgroups normal.
type: Item
instance of: scholarly article
title: Multiplicative Jordan decomposition in group rings and \(p\)-groups with all noncyclic subgroups normal.
zbMATH Open document ID: 1276.16021
author: Chia-Hsin Liu
published in: Journal of Algebra
publication date: 2013-05-27T00:00:00Z
review text: Let \(G\) be a finite \(p\)-group, \(\mathbb ZG\) its integral group ring. A unit in the rational group algebra \(\mathbb QG\) always has a multiplicative Jordan decomposition (MJD for short) i.e. it is a product of a semisimple unit and a commuting unipotent unit. \(G\) is said to have the property MJD if every unit of \(\mathbb ZG\) has its semisimple and unipotent factors in \(\mathbb ZG\). Research on MJD was initiated by \textit{S. R. Arora, A. W. Hales} and \textit{I. B. S. Passi} [J. Algebra 209, No. 2, 533-542 (1998; Zbl 0919.16021)], and in \textit{A. W. Hales, I. B. S. Passi} and \textit{L. E. Wilson} [J. Algebra 316, No. 1, 109-132 (2007); Corrigendum 371, 665-666 (2012; Zbl 1144.16020)] 2-groups \(G\) with MJD were completely determined. Based upon the results of \textit{C.-H. Liu} and \textit{D. S. Passman} [in J. Algebra Appl. 8, No. 4, 505-519 (2009; Zbl 1185.20002) and ibid. 9, No. 3, 483-492 (2010; Zbl 1205.16019)], and the complete classification of groups \(G\) with every noncyclic subgroup normal by \textit{Z. Božikov} and \textit{Z. Janko} [Glas. Mat., III. Ser. 44, No. 1, 177-185 (2009; Zbl 1194.20016)], the author reproves the determination of 2-groups \(G\) with MJD in a different way, and for odd \(p\) establishes that for the group \(G\) with MJD it is necessary to have each of its noncyclic subgroups normal.
reviewed by: János Kurdics
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 6166660
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: multiplicative Jordan decompositions
zbMATH Keywords: additive Jordan decompositions
zbMATH Keywords: integral group rings
zbMATH Keywords: units
zbMATH Keywords: semisimple elements
zbMATH Keywords: unipotent elements
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2012.07.010
OpenAlex ID: W2070653756
cites work: The multiplicative Jordan decomposition in group rings
cites work: A complete classification of finite p-groups all of whose noncyclic subgroups are normal
cites work: The multiplicative Jordan decomposition in group rings. II.
cites work: 2-groups with normal noncyclic subgroups
cites work: MULTIPLICATIVE JORDAN DECOMPOSITION IN GROUP RINGS OF 3-GROUPS
cites work: MULTIPLICATIVE JORDAN DECOMPOSITION IN GROUP RINGS OF 2, 3-GROUPS
cites work: MULTIPLICATIVE JORDAN DECOMPOSITION IN INTEGRAL GROUP RINGS OF GROUPS OF ORDER 16
cites work: Nonnormal subgroups of \(p\)-groups.
DOI: 10.1016/J.JALGEBRA.2012.07.010
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description: scientific article; zbMATH DE number 6166660