## 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

### Citations:

Zbl 0784.16020; Zbl 0788.16024; Zbl 0912.16013; Zbl 0968.16015
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### References:

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