## 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)].

### 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 0788.16024; Zbl 1034.16037
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### References:

 [1] Arora S. R., Comm. Algebra 21 pp 3673– (1993) · Zbl 0788.16024 [2] Arora S. R., Comm. Algebra 21 pp 25– (1993) · Zbl 0784.16020 [3] Jespers E., J. Algebra 247 pp 24– (2002) · Zbl 1063.16036 [4] Li Y., Canadian Journal of Mathematics 50 pp 401– (1998) · Zbl 0912.16013 [5] Li Y., Bull. Austral. Math. Soc. 67 pp 171– (2003) · Zbl 1026.16019 [6] Li Y., Proc. Amer. Math. Soc. 129 pp 2235– (2001) · Zbl 0968.16015 [7] Li Y., Comm. Algebra 31 pp 3207– (2003) · Zbl 1034.16037 [8] Sehgal S. K., Units in Integral Group Rings (1993) · Zbl 0803.16022
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