Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-15T19:31:08.875Z Has data issue: false hasContentIssue false
  • English
  • Français

Representations of infinite soluble groups

Published online by Cambridge University Press:  18 May 2009

Ian M. Musson
Ian M. Musson
Affiliation:
University of Wisconsin-Madison Madison, WI 53706USA
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The purpose of this paper is to study the following two questions.

(1) When does the group algebra of a soluble group have infinite dimensional irreducible modules?

(2) When is the group algebra of a torsion free soluble group primitive?

In relation to the first question, Roseblade [13] has proved that if G is a polycyclic group and k an absolute field then all irreducible kG-modules are finite dimensional. Here we prove a converse.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 24 , Issue 1 , January 1983 , pp. 43 - 52
Copyright
Copyright © Glasgow Mathematical Journal Trust 1983

References

REFERENCES

1.Bergman, G. M., The logarithmic limit set of an algebraic variety, Trans. Amer. Math. Soc. 157 (1971), 459–469.Google Scholar
2.Brown, K. A., Primitive group rings of soluble groups, Arch. Math. (Basel) 36 (1981), 404–413.Google Scholar
3.Farkas, D. R., Group rings, an annotated questionaire, Comm. Alg. 8 (1980), 585–602.Google Scholar
4.Farkas, D. R. and Passman, D. S., Primitive Noetherian group rings, Comm. Alg. 5 (1978), 301–315.Google Scholar
5.Farkas, D. R. and Snider, R. L., Group algebras whose simple modules are injective, Trans. Amer. Math. Soc. 194 (1974), 241–248.Google Scholar
6.Fuchs, L., Abelian groups (Pergamon, 1960).Google Scholar
7.Hartley, B., Injective modules over group rings, Quart. J. Math. 28 (1977), 1–29.Google Scholar
8.Irving, R. S., Some more primitive group rings, Israel J. Math. 37 (1980), 331–350.CrossRefGoogle Scholar
9.Jategaonkar, A. V., Integral group rings of polycyclic-by-finite groups, J. Pure and Applied Algebra 4 (1974), 337–343.CrossRefGoogle Scholar
10.Kaplansky, I., Commutative rings (University of Chicago Press, 1974).Google Scholar
11.Passman, D. S., The algebraic structure of group rings(Wiley-Interscience, 1977).Google Scholar
12.Robinson, D. J. S., Finiteness conditions and generalized soluble groups, parts 1 and 2 (Springer-Verlag, 1972).Google Scholar
13.Roseblade, J. E., Group rings of polycyclic groups, J. Pure and Applied Algebra 3 (1974), 307–328.Google Scholar
14.Roseblade, J. E., Prime ideals in group rings of polycyclic groups, Proc. London Math. Soc. 36 (1978), 385–447.CrossRefGoogle Scholar
15.Snider, R. L., Soluble groups whose representations are finite dimensional, Abstracts Amer. Math. Soc.(1980), Abstract 775–A6, page 250.Google Scholar
16.Wehrfritz, B. A. F., Infinite linear groups (Springer-Verlag, 1973).Google Scholar
17.Wehrfritz, B. A. F., Groups whose irreducible representations have finite degree I, Math. Proc. Cambridge Philos. Soc. 90 (1981), 411–421.CrossRefGoogle Scholar