Geodesic
Free Subgroups in the Unit Groups of Integral Group Rings
Canadian journal of mathematics, Tome 32 (1980) no. 6, pp. 1342-1352

Voir la notice de l'article provenant de la source Cambridge University Press

Let G be a group, Z G the group ring of G over the ring Z of integers, and U(Z G) the group of units of Z G. One method of investigating U(Z G) is to choose some property of groups and try to determine the groups G such that U(Z G) enjoys that property. For example Sehgal and Zassenhaus [9] have given necessary and sufficient conditions for U(Z G) to be nilpotent (see also [7]), and the same authors have investigated when U(Z G) is an FC (finite-conjugate) group [10]. For a survey of related questions, see [3]. In this paper we consider when U(Z G) 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). 
Hartley, B.; Pickel, P. F. Free Subgroups in the Unit Groups of Integral Group Rings. Canadian journal of mathematics, Tome 32 (1980) no. 6, pp. 1342-1352. doi: 10.4153/CJM-1980-104-3
@article{10_4153_CJM_1980_104_3,
     author = {Hartley, B. and Pickel, P. F.},
     title = {Free {Subgroups} in the {Unit} {Groups} of {Integral} {Group} {Rings}},
     journal = {Canadian journal of mathematics},
     pages = {1342--1352},
     year = {1980},
     volume = {32},
     number = {6},
     doi = {10.4153/CJM-1980-104-3},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1980-104-3/}
}
TY  - JOUR
AU  - Hartley, B.
AU  - Pickel, P. F.
TI  - Free Subgroups in the Unit Groups of Integral Group Rings
JO  - Canadian journal of mathematics
PY  - 1980
SP  - 1342
EP  - 1352
VL  - 32
IS  - 6
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1980-104-3/
DO  - 10.4153/CJM-1980-104-3
ID  - 10_4153_CJM_1980_104_3
ER  - 
%0 Journal Article
%A Hartley, B.
%A Pickel, P. F.
%T Free Subgroups in the Unit Groups of Integral Group Rings
%J Canadian journal of mathematics
%D 1980
%P 1342-1352
%V 32
%N 6
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1980-104-3/
%R 10.4153/CJM-1980-104-3
%F 10_4153_CJM_1980_104_3

[1] 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] 2. Brenner, J. L., Quelques groupes libres de matrices, C.R. Acad. Sci. Paris 241 (1955), 1689–1691. Google Scholar

[3] 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] 4. Hartley, B., Finite groups of automorphisms of locally soluble groups, J. Algebra 57 (1979), 242–257. Google Scholar

[5] 5. Higman, Graham, The units of group rings, Proc. London. Math. Soc. 57 (1940), 231–248. Google Scholar

[6] 6. Kuros, A. G., Theory of groups, Vol. II, trans. Hirsch, K. A. (Chelsea, New York, 1956). Google Scholar

[7] 7. Polcino, Milies, C., Integral group rings with nilpotent unit groups, to appear. Google Scholar

[8] 8. Schenkman, E., Group theory (van Nostrand, Princeton, 1965). Google Scholar

[9] 9. Sehgal, S. K. and Zassenhaus, H., Integral group rings with nilpotent unit groups, Comm. Algebra 5 (1977), 101–111. Google Scholar

[10] 10. Sehgal, S. K. and Zassenhaus, H., Group rings whose units form an FC-group, Math. Z. 153 (1977), 29–35. Google Scholar

[11] 11. Sehgal, S. K., Topics in group rings (Marcel Dekker, New York, 1978). Google Scholar

[12] 12. Tits, J., Free subgroups in linear groups, J. Algebra 20 (1972), 250–270. Google Scholar

Cité par Sources :