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Free Subgroups in the Unit Groups of Integral Group Rings

Published online by Cambridge University Press:  20 November 2018

B. Hartley  and
P. F. Pickel
B. Hartley
Affiliation:
University of Manchester, Manchester, England
P. F. Pickel
Affiliation:
Polytechnic Institute of New York, Farmingdale, New York
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Let G be a group, ZG the group ring of G over the ring Z of integers, and U(ZG) the group of units of ZG. One method of investigating U(ZG) is to choose some property of groups and try to determine the groups G such that U(ZG) enjoys that property. For example Sehgal and Zassenhaus [9] have given necessary and sufficient conditions for U(ZG) to be nilpotent (see also [7]), and the same authors have investigated when U(ZG) is an FC (finite-conjugate) group [10]. For a survey of related questions, see [3]. In this paper we consider when U(ZG) contains a free subgroup of rank 2. We conjecture that if this does not happen, then every finite subgroup of G is normal, from which various other conclusions then follow (see Lemma 4).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 6 , 01 December 1980 , pp. 1342 - 1352
Copyright
Copyright © Canadian Mathematical Society 1980

References

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