The upper central series of the unit group of an integral group ring. (English) Zbl 1078.16029

Let \(Z_n(U)\) denote the \(n\)-th term of the upper central series of the unit group \(U=U(\mathbb{Z} G)\) of the integral group ring \(\mathbb{Z} G\) and \(\widetilde Z=\bigcup^\infty_{i=1}Z_n(U)\). The authors show that if the set of the torsion elements of \(G\) forms a subgroup \(T\) and \(\widetilde Z\not\subset C_U(T)\), then \(T\) is either an Abelian 2-group or a \(Q\)-group [for the definition of \(Q\)-group see S. R. Arora and I. B. S. Passi, Commun. Algebra 21, No. 10, 3673-3683 (1993; Zbl 0788.16024)]. Moreover, if \(T\) is a subgroup of \(G\) and \(\widetilde Z\subseteq N_U(G)\), then \(\widetilde Z\subseteq G\cdot C_U(T)\). Recall that if \(G\) is an FC-group, then the property \(\widetilde Z\subseteq N_U(G)\) holds [see the authors, Commun. Algebra 31, No. 7, 3207-3217 (2003; Zbl 1034.16037)].


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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