## 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

### Citations:

Zbl 0046.02202; Zbl 0736.20017; Zbl 0840.16027; Zbl 0784.16020
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