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Some results on hypercentral units in integral group rings. (English) Zbl 1034.16037

Let \(G\) be a group and let \(Z_n(G)\) denote the \(n\)-th term of its upper central series, \(\widetilde Z(G)=\bigcup_nZ_n(G)\) its hypercentre, \(t(G)\) the set of its torsion elements and, for \(S\subseteq G\), \(C_G(S)\) the centralizer of \(S\) in \(G\). Let \(\mathcal U=\mathcal U(\mathbb{Z}[G])\) be the group of units of the integral group ring \(\mathbb{Z}[G]\) of \(G\). If \(G\) is finite, then S. R. Arora, A. W. Hales and the reviewer [Commun. Algebra 21, No. 1, 25-35 (1993; Zbl 0784.16020)] proved that \(\widetilde Z(\mathcal U)=Z_2(\mathcal U)\) and the hypercentral torsion units are contained in \(\pm Z_2(G)\). S. R. Arora and the reviewer [ibid. 21, No. 10, 3673-3683 (1993; Zbl 0788.16024)] proved that \(Z_2(\mathcal U)=t(\widetilde Z(\mathcal U))Z_1(\mathcal U)\), and so \(\widetilde Z(\mathcal U)\subseteq G\cdot Z_1(\mathcal U)\). These results were extended to torsion groups by the first author [Can. J. Math. 50, No. 2, 401-411 (1998; Zbl 0912.16013)] and the authors [Proc. Am. Math. Soc. 129, No. 8, 2235-2238 (2001; Zbl 0968.16015)].
Continuing this work the authors investigate the inclusions (i) \(\widetilde Z(\mathcal U)\subseteq G\cdot Z_1(\mathcal U)\) and (ii) \(\widetilde Z(\mathcal U)\subseteq G\cdot C_{\mathcal U}(t(G))\) for arbitrary groups \(G\). Various classes of groups are given for which (i) holds. Towards (ii) it is proved that if \(T:=t(G)\) constitutes an Abelian subgroup of \(G\) and \(\mathbb{Z}[G/T]\) has only trivial units, then (ii) holds; a more precise description of the terms \(Z_n(\mathcal U)\) of the upper central series is provided if, in addition, \(G/T\) is cyclic and every finite subgroup of \(G\) is normal in \(G\). The main result of the paper is a characterization of \(T\) in case it is a subgroup and \(Z_2(\mathcal U)\not\subseteq C_{\mathcal U}(T)\).

MSC:

16U60 Units, groups of units (associative rings and algebras)
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
16S34 Group rings
20F14 Derived series, central series, and generalizations for groups
20E07 Subgroup theorems; subgroup growth
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References:

[1] DOI: 10.1080/00927879308824756 · Zbl 0788.16024
[2] DOI: 10.1080/00927879208824548 · Zbl 0784.16020
[3] DOI: 10.2307/3062112 · Zbl 0990.20002
[4] DOI: 10.1006/jabr.2001.8724 · Zbl 1063.16036
[5] DOI: 10.4153/CJM-1998-021-2 · Zbl 0912.16013
[6] DOI: 10.1016/S0021-8693(02)00102-3 · Zbl 1017.16023
[7] DOI: 10.1090/S0002-9939-01-05848-8 · Zbl 0968.16015
[8] Sehgal S. K., Topics in Group Rings (1978) · Zbl 0411.16004
[9] Sehgal S. K., Units in Integral Group Rings (1993) · Zbl 0803.16022
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