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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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[1] DOI: 10.1080/00927879208824548 · Zbl 0784.16020 · doi:10.1080/00927879208824548
[2] DOI: 10.1016/0021-8693(66)90018-4 · Zbl 0141.02401 · doi:10.1016/0021-8693(66)90018-4
[3] Curtis C.W., Methods of Representation Theory 1 (1981)
[4] DOI: 10.1090/S0002-9939-1990-0994785-7 · doi:10.1090/S0002-9939-1990-0994785-7
[5] Robinson Derek J.S., A Course in the Theory of Groups (1982) · Zbl 0483.20001
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