Multiplicative Jordan decomposition in group rings and \(p\)-groups with all noncyclic subgroups normal.
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.
scientific article; zbMATH DE number 6166660
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