Factor morphisms of nilpotent groups

Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 32, No. 3, pp. 119–125, May–June, 1991.     

Some nilpotent and locally nilpotent matrix groups

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.     

Representations of nilpotent locally compact groups

The main result of this paper is that if all the σ-representations of a separable locally compact nilpotent group are type I, then all the irreducible σ-representations are (up to equivalence) induced from one-dimensional σ-representations. This generalizes a result of Kirillov for simply connected nilpotent Lie groups. Parallels with the orbit theory for Lie groups are drawn.     

On the existence of stably nilpotent subgroups in locally nilpotent groups

A condition for the existence of stably nilpotent subgroups of a given nilpotency class in a locally nilpotent group is established.Translated from Matematicheskie Zametki, Vol. 4, No. 3, pp. 361–368, September, 1968.     

Centralizers of finite nilpotent subgroups in locally finite groups

We investigate the structure of locally finite groups with a finite subgroup whose centralizer is close to a linear group. Deceased. Translated fromAlgebra i Logika, Vol. 35 No. 4, pp. 389–410, July–August, 1996.