Some nilpotent and locally nilpotent matrix groups

Abstract

Say a division ring D is special if for every finite subset X of D there is a homomorphism of the subring of D generated by X into a division ring of finite Schur index a power of its positive characteristic. ( D is not assumed to have positive characteristic.) We make a detailed study of nilpotent and locally nilpotent matrix groups over special division rings. This has been done previously for a number of ‘special’ division rings arising from group algebras and Lie algebras, particularly by A.I. Lichtman. The present paper therefore presents single proofs of all these results. It also covers many division rings not considered before and produces some new results for those that have been considered before. In view of the definition of ‘special’ it is not surprising that the proofs depend on a detailed analysis of the finite-dimensional case.