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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

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