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Central height of the unit group of an integral group ring. (English) Zbl 0788.16024
Let $$G$$ be a finite group and let $$V=V(ZG)$$ be the group of normalized units of its integral group ring $$ZG$$. It is proved by A. W. Hales and the authors [Commun. Algebra 21, 25-35 (1993; Zbl 0784.16020)] that the central height of $$V$$ is at most 2, i.e. $$Z_ 2(V)=Z_ 3(V)$$, where $$\{Z_ i(V)\}$$ denotes the upper central series of $$V$$. In this paper the authors prove that $$Z_ 2(V)=Z_ 1(V)T$$, where $$T$$ denotes the torsion subgroup of $$Z_ 2(V)$$. This yields a characterization of the unit groups with central height 2. In view of the work of J. Ritter and S. K. Sehgal [Proc. Am. Math. Soc. 108, 327-329 (1990; Zbl 0688.16009)], which characterizes the unit groups with central height 0, this result completes the classification of the admissible values of the central height.

##### MSC:
 16U60 Units, groups of units (associative rings and algebras) 16S34 Group rings 20C05 Group rings of finite groups and their modules (group-theoretic aspects) 20F14 Derived series, central series, and generalizations for groups
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##### References:
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