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Free Subgroups and the Residual Nilpotence of the Group of Units of Modular and p-Adic Group Rings

Published online by Cambridge University Press:  20 November 2018

Jairo Zacarias Gonçalves
Jairo Zacarias Gonçalves*
Affiliation:
Universidade de Sāo Paulo, Instituto de Mat. E EstatísticaAG. Iguatemi, CX. Postal 20570 01000 Sāo Paulo, SP., Brazil
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Abstract

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Let G be a group, let RG be the group ring of the group G over the unital commutative ring R and let U(RG) be its group of units. Conditions which imply that U(RG) contains no free noncyclic group are studied, when R is a field of characteristic p ≠ 0, not algebraic over its prime field, and G is a solvable-by-finite group without p-elements. We also consider the case R = ℤp, the ring of p-adic integers and G torsionby- nilpotent torsion free group. Finally, the residual nilpotence of U(ℤpG) is investigated.

Keywords

16A2616A4020CO5group ringsgroup of unitsfree groupsresidual nilpotence
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 29 , Issue 3 , 01 September 1986 , pp. 321 - 328
Copyright
Copyright © Canadian Mathematical Society 1986

References

1. Gonçalves, J. Z., Group rings with solvable unit groups. Comm. in Algebra, 14, 1 (1986), pp. 1—20.Google Scholar
2. Gonçalves, J. Z., Free subgroups of units in group rings. Bull Can. Math. Soc. 27, 3 (1984), pp. 309–312.Google Scholar
3. Gonçalves, J. Z., Free subgroups in the group of units of group rings II. Journal of Number Theory, 21, 2 (1985), pp. 121–127.Google Scholar
4. Gonçalves, J. Z., Free groups in subnormal subgroups and the residual nilpotence of the group of units of group rings. Bull Can. Math. Soc. 27, 3 (1984), pp. 365–370.Google Scholar
5. Hartley, B. and Pickel, P. F., Free subgroups in the unit groups of integral group rings. Can. J. of Math. 32, 6(1980), pp. 1342–1352.Google Scholar
6. Musson, I., and Weiss, A., Integral group rings with residually nilpotent unit groups. Arch. Math. 38 (1982), pp. 514–530.Google Scholar
7. Sehgal, S. K., Topics in group rings. Marcel Dekker, New York, 1978.Google Scholar
8. Sehgal, S. K. and Zassenhaus, H.J., Isomorphism of integral group rings of abelian-by-nilpotent class two groups. Pre-print.Google Scholar
9. Tits, J., Free subgroups in linear groups. J of Algebra 20 (1972), pp. 250—270.Google Scholar
10. Warhurst, D. S., Topics in group rings. Thesis. University of Manchester, 1981.Google Scholar
11. Wehrfritz, B. A. F., Infinite linear groups. Springer Verlag, Berlin, 1973.Google Scholar