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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.

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
Full Text: DOI
[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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