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Some matrix groups over finite-dimensional division algebras

Published online by Cambridge University Press:  20 January 2009

B. A. F. Wehrfritz
B. A. F. Wehrfritz
Affiliation:
Queen Mary College, London El 4NS
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Abstract

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Let n be a positive integer and D a division algebra of finite dimension m over its centre. We describe in detail the structure of a soluble subgroup G of GL(n,D). (More generally we consider subgroups of GL{n,D) with no free subgroup of rank 2.) Of course G is isomorphic to a linear group of degree mn and hence linear theory describes G, but the object here is to reduce as far as possible the dependence of the description on m. The results are particularly sharp if n=l. They will be used in later papers to study matrix groups over certain types of infinite-dimensional division algebra. This present paper was very much inspired by A. I. Lichtman's work: Free subgroups in linear groups over some skew fields, J. Algebra105 (1987), 1–28.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 33 , Issue 1 , February 1990 , pp. 97 - 111
Copyright
Copyright © Edinburgh Mathematical Society 1990

References

REFERENCES

1.Acobson, N. J., Lie Algebras (John Wiley & Sons, New York, 1962).Google Scholar
2.Jacobson, N., Basic Algebra II (W. H. Freeman, San Francisco, 1980).Google Scholar
3.Lichtman, A. I., Free subgroups in linear groups over some skew fields, J. Algebra 105 (1987), 1–28.CrossRefGoogle Scholar
4.Shirvani, M. and Wehrfritz, B. A. F., Skew Linear Groups (Cambridge University Press, Cambridge, 1986).Google Scholar
5.Tits, J., Free subgroups in linear groups, J. Algebra 20 (1972), 250–270.CrossRefGoogle Scholar
6.Wehrfritz, B. A. F., Infinite Linear Groups (Springer-Verlag, Berlin, 1973).CrossRefGoogle Scholar
7.Wehrfritz, B. A. F., Some skew linear groups associated with Lie algebras of characteristic zero, J. Algebra 121 (1989), 155–168.CrossRefGoogle Scholar
8.Wehrfritz, B. A. F., Some nilpotent and locally nilpotent matrix groups, J. Pure Appl. Algebra, to appear.Google Scholar