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Irreducible locally nilpotent finitary skew linear groups

Published online by Cambridge University Press:  20 January 2009

B. A. F. Wehrfritz
B. A. F. Wehrfritz
Affiliation:
School of Mathematical SciencesQueen Mary and Westerfield CollegeMile End RoadLondon E1 4NSEngland
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Abstract

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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
Information
Proceedings of the Edinburgh Mathematical Society , Volume 38 , Issue 1 , February 1995 , pp. 63 - 76
Copyright
Copyright © Edinburgh Mathematical Society 1995

References

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