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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

1. Berman, S. D., On the equation xm = 1 in an integral group ring, Ukrain. Mat. Z. 7 (1955), 253–261 (Russian).Google Scholar
2. Brenner, J. L., Quelques groupes libres de matrices, C.R. Acad. Sci. Paris 241 (1955), 1689–1691.Google Scholar
3. Dennis, R. K., The structure of the unit group of group rings, Proc. Ring Theory Conference, University of Oklahoma (Marcel Dekker, New York, 1976).Google Scholar
4. Hartley, B., Finite groups of automorphisms of locally soluble groups, J. Algebra 57 (1979), 242–257.Google Scholar
5. Higman, Graham, The units of group rings, Proc. London. Math. Soc. 57 (1940), 231–248.Google Scholar
6. Kuros, A. G., Theory of groups, Vol. II, trans. Hirsch, K. A. (Chelsea, New York, 1956).Google Scholar
7. Polcino, Milies, C., Integral group rings with nilpotent unit groups, to appear.Google Scholar
8. Schenkman, E., Group theory (van Nostrand, Princeton, 1965).Google Scholar
9. Sehgal, S. K. and Zassenhaus, H., Integral group rings with nilpotent unit groups, Comm. Algebra 5 (1977), 101–111.Google Scholar
10. Sehgal, S. K. and Zassenhaus, H., Group rings whose units form an FC-group, Math. Z. 153 (1977), 29–35.Google Scholar
11. Sehgal, S. K., Topics in group rings (Marcel Dekker, New York, 1978).Google Scholar
12. Tits, J., Free subgroups in linear groups, J. Algebra 20 (1972), 250–270.Google Scholar