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Free Groups in Subnormal Subgroups and the Residual Nilpotence of the Group of Units of Groups Rings

Published online by Cambridge University Press:  20 November 2018

Jairo Z. Gonçalves
Jairo Z. Gonçalves*
Affiliation:
Instituto de Matemática e Estatstica, Universidade de Săo Paulo, Brasil
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Abstract

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Let KG be the group ring of the group G over the field K and U(KG) its unit group. When G is finite we derive conditions which imply that every noncentral subnormal subgroup of U(KG) contains a free group of rank two. We also show that residual nilpotence of U(KG) coincides with nilpotence, this being no longer true if G is infinite.

We can answer partially the following question: when is G sub-normal in U(KG)?

Keywords

16A2616A4020C0520E0520E25
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 27 , Issue 3 , 01 September 1984 , pp. 365 - 370
Copyright
Copyright © Canadian Mathematical Society 1984

References

1. Bateman, J. M. and Coleman, D. B., Group algebras with nilpotent unit groups. Proc. Amer. Math. Soc. 19, 2 (1968), 448–449.Google Scholar
2. Gonçalves, J. Z., Free subgroups of units in group rings. Can. Math. Bulletin (to appear).Google Scholar
3. Hartley, B. and Pickel, P. F., Free subgroups in the unit groups of integral group rings. Can. J. of Math. 32, 6 (1980), 1342–1352.Google Scholar
4. Herstein, I. N., Multiplicative commutators in division rings Israel J. of Math. 31, 2 (1978), 180–188.Google Scholar
5. Lichtman, A. I., On subgroups of the multiplicative group of skew fields. Proc. Amer. Math. Soc. 63, 1 (1977), 15–16.Google Scholar
6. Musson, I. and Weiss, A., Integral group rings with residually nilpotent unit groups. Arch. Math. 38 (1982), 514–530.Google Scholar
7. Polcino Milies, C., Units of group rings: a short survey. Groups—St. Andrews, 1981, London Math. Soc. Lee. Notes Series, 71, London, 1982, 281–297.Google Scholar
8. Sehgal, S. K., Nilpotent elements in group rings. Manuscripta Math. 15 (1975), 65–80.Google Scholar
9. Sehgal, S. K., Topics in group rings. Marcel Dekker, New York, 1978.Google Scholar
10. Scott, W. R., Group Theory. Prentice-Hall, Englewood Cliffs, N. Jersey, 1964.Google Scholar
11. Suprunenko, D. A., Matrix groups. Translations of Mathematics Monographs, vol 45. Amer. Math. Soc. Providence, Rhode Island, 1976.Google Scholar
12. Tits, J., Free subgroups in linear groups. J. of Algebra, 20 (1972), 250–270.Google Scholar