Induced Representations of Abelian Groups of Finite Rank

We prove that any irreducible faithful representation of an almost torsion-free Abelian group G of finite rank over a finitely generated field of characteristic zero is induced from an irreducible representation of a finitely generated subgroup of the group G.     

On ideals of the group algebra of a free group of degree of freedom two over the field of complex numbers

In this paper, we prove the existence of an element of the group algebra A=F of a free groupF with two generatorsx andy over the field of complex numbersC such that, for any complexa andb for which ¦a¦=¦b¦=1, we haveA _a,b ()A=0, where _a,b ( is an automorphism ofA that mapsx,y intoax, by, respectively. Thus, we give a negative answer to question 12.46 of P. A. Linnel from Kourovka Notebook.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No.4, pp. 571–572, April, 1995.     

On solvable groups,all proper factor groups of which have finite ranks

This paper deals with finitely generated finitely approximable solvable groups of infinite special rank, all proper normal subgroups of which determine the factor groups of finite special ranks.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 9, pp. 1274–1281, September, 1993.     

On Solvable Groups with Proper Quotient Groups of Finite Rank

We study solvable groups of infinite special rank all proper normal subgroups of which define quotient groups of finite special rank.     

On Primitive Representations of Minimax Nilpotent Groups

We show, in particular, that in the class of minimax two-step nilpotent groups only finitely generated groups can admit exact irreducible primitive representations over a finitely generated field of characteristic zero. We also suggest some approaches to studying induced representations of nilpotent groups of finite rank.     

On Noetherian Modules over Minimax Abelian Groups

We consider modules over minimax Abelian groups. We prove that if A is an Abelian minimax subgroup of the multiplicative group of a field k and if the subring K of the field k generated by the subgroup A is Noetherian, then the subgroup A is the direct product of a periodic group and a finitely generated group.     

Noether modules over abelian groups of finite free rank

We prove that if M is a Noether JG-module, where G is an abelian group of finite free rank, and either J=, or J=Ft, where F is a finite field and t is an infinite cyclic group, then the module M belongs to a class (J, ) for some finite set in the sense defined by P. Hall.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 1042–1048, July–August, 1991.