Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-24T02:27:34.287Z Has data issue: false hasContentIssue false

Irreducible locally nilpotent finitary skew linear groups

Published online by Cambridge University Press:  20 January 2009

B. A. F. Wehrfritz
Affiliation:
School of Mathematical SciencesQueen Mary and Westerfield CollegeMile End RoadLondon E1 4NSEngland
Rights & Permissions [Opens in a new window]

Abstract

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.

Let V be a left vector space over the arbitrary division ring D and G a locally nilpotent group of finitary automorphisms of V (automorphisms g of V such that dimDV(g-1)<∞) such that V is irreducible as D-G bimodule. If V is infinite dimensional we show that such groups are very rare, much rarer than in the finite-dimensional case. For example we show that if dimDV is infinite then dimDV = |G| = ℵ0 and G is a locally finite q-group for some prime q ≠ char D. Moreover G is isomorphic to a finitary linear group over a field. Examples show that infinite-dimensional such groups G do exist. Note also that there exist examples of finite-dimensional such groups G that are not isomorphic to any finitary linear group over a field. Generally the finite-dimensional examples are more varied.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 1995

References

REFERENCES

1.Kegel, O. H., and Wehrfritz, B. A. F., Locally Finite Groups (North-Holland Pub. Co., Amsterdam, 1973).Google Scholar
2.Kropholler, P. H., Linnell, P. A. and Moody, J. A., Applications of a new K-theoretic theorem to soluble group rings, Proc. Amer. Math. Soc. 104 (1988), 675684.Google Scholar
3.Lichtman, A. I. and Wehrfritz, B. A. F., Finite dimensional subalgebras in matrix rings over transcendental division algebras, Proc. Amer. Math. Soc. 106 (1989), 335344.Google Scholar
4.Mcconnell, J. C. and Robson, J. C., Non-commutative Noetherian Rings (John Wiley & Sons, Chichester, 1987).Google Scholar
5.Neumann, P. M., The lawlessness of groups of finitary permutations, Arch. Math. 26 (1975), 561566.CrossRefGoogle Scholar
6.Robinson, D. J. S., Finiteness Conditions and Generalized Soluble Groups Vol 2 (Springer-Verlag, Berlin, 1972).CrossRefGoogle Scholar
7.Shirvani, M. and Wehrfritz, B. A. F., Skew Linear Groups (Cambridge Univ. Press, Cambridge, 1986).Google Scholar
8.Wehrfritz, B. A. F., Infinite Linear Groups (Springer-Verlag, Berlin, 1973).CrossRefGoogle Scholar
9.Wehrfritz, B. A. F., Soluble and locally soluble skew linear groups, Arch. Math. 49 (1987), 379388.CrossRefGoogle Scholar
10.Wehrfritz, B. A. F., Some nilpotent and locally nilpotent matrix groups, J. Pure Appl. Algebra 60 (1989), 289312.CrossRefGoogle Scholar
11.Wehrfritz, B. A. F., Locally soluble finitary skew linear groups, J. Algebra 60 (1993) 226241.CrossRefGoogle Scholar
12.Wehrfritz, B. A. F., Algebras generated by locally nilpotent finitary skew linear groups, J. Pure App. Algebra 88 (1993), 305316.CrossRefGoogle Scholar
13.Wehrfritz, B. A. F., Locally nilpotent finitary skew linear groups, J. London Math. Soc., to appear.Google Scholar