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The hypercentre and the \(n\)-centre of the unit group of an integral group ring. (English) Zbl 0912.16013

The paper concerns the \(n\)-centre \(Z(U,n)\) of the unit group \(U\) of an integral group ring \(\mathbb{Z} G\) with \(G\) periodic. The definition of \(Z(U,n)\), for any group \(U\), goes back to R. Baer [Math. Ann. 124, 161-177 (1952; Zbl 0046.02202)]; \(Z(U,n)=\{u\in U:(uv)^n=u^nv^n\;(\forall v\in U)\}\). Earlier work in this respect (which the paper is based on) is due to, mainly, L. C. Kappe and M. L. Newell [Lond. Math. Soc. Lect. Note Ser. 160, 339-352 (1991; Zbl 0736.20017)], and M. M. Parmenter [Commun. Algebra 23, No. 14, 5503-5507 (1995; Zbl 0840.16027)]. Moreover, the theory of group ring units is quite extensively exploited.
Theorem A: In the upper central series of \(U\), \(1=Z_0(U)\leq Z_1(U)\leq\ldots\), equality holds at 2, i.e. \(Z_2(U)=Z_3(U)\). For \(G\) finite, this restates a result of S. R. Arora, A. W. Hales and I. B. S. Passi [Commun. Algebra 21, No. 1, 25-35 (1993; Zbl 0784.16020)]. Theorem B: \(Z(U,3)=Z(U,2)=Z_1(U)\). Theorem C: \(Z(U,n)=U\) or \(Z_2(U)\) or \(Z_1(U)\) according to \(n=0,1\) or \(n=4k\), \(4k+1\) (\(k\geq 1\)) or \(n=4k+2\), \(4k+3\) (\(k\geq 0\)).

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
20F50 Periodic groups; locally finite groups
20F14 Derived series, central series, and generalizations for groups
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