Locally nilpotent groups of finite non-Abelian sectional rank

We introduce the notion of non-Abelian sectional rank of a group and study locally nilpotent non-Abelian groups of finite non-Abelian sectional rank. It is proved that the (special) rank of these groups is finite.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 4, pp. 452–455, April, 1995.     

Locally nilpotent groups with a centralizer of finite rank

Locally nilpotent groups in which the centralizer of some finitely generated subgroup has finite rank are studied. It is shown that if G is such a group and F is a finitely generated subgroup with centralizer C_G(F) of finite rank, then the centralizer of the image of F in the factor group G/t(G) modulo the periodic part t(G) also has finite rank. It is also shown that G is hypercentral when F is cyclic and either G is torsion-free or all Sylow subgroups of the periodic part of C_G(F) are finite.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 11, pp. 1511–1517, November, 1992.     

On the lower central series of the free nilpotent groups of finite rank

In their article, “On the derived subgroup of the free nilpotent groups of finite rank” R. D. Blyth, P. Moravec, and R. F. Morse describe the structure of the derived subgroup of a free nilpotent group of finite rank n as a direct product of a nonabelian group and a free abelian group, each with a minimal generating set of cardinality that is a given function of n. They apply this result to computing the nonabelian tensor squares of the free nilpotent groups of finite rank. We generalize their main result to investigate the structure of the other terms of the lower central series of a free nilpotent group of finite rank, each again described as a direct product of a nonabelian group and a free abelian group. In order to compute the ranks of the free abelian components and the size of minimal generating sets for the nonabelian components we introduce what we call weight partitions.     

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.     

Products of finite nilpotent groups

Dedicated to Professor H. Wielandt on the occasion of his eightieth birthday     

Modules over group rings of solvable groups with rank restrictions on subgroups

Let A be an R G-module over a commutative ring R, where G is a group of infinite section p-rank (0-rank), C _G (A) = 1, A is not a Noetherian R-module, and the quotient A/C _A (H) is a Noetherian R-module for every proper subgroup H of infinite section p-rank (0-rank). We describe the structure of solvable groups G of this type.     

2-step nilpotent Lie groups of higher rank

We construct a family of simply connected 2-step nilpotent Lie groups of higher rank such that every geodesic lies in a flat. These are as Riemannian manifolds irreducible and arise from real representations of compact Lie algebras. Moreover we show that groups of Heisenberg type do not even infinitesimally have higher rank. Received: 2 July 2001 / Revised version: 19 October 2001     

Induced modules over group algebras of metabelian groups of finite rank

ABSTRACT

Investigation of multiplace functions by algebraic methods plays an important role in modern mathematics were we consider various operations on sets of functions, which are naturally defined. The basic operation for functions is superposition (composition), but there are some other naturally defined operations, which are also worth of consideration. For example, the operation of set-theoretic intersection and the operation of projections. In this paper we find an abstract characterization of the set of multiplace functions which are closely related to these three operations.     

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.     

Large nilpotent subgroups of finite simple groups

Orders and the structure of large nilpotent subgroups in all finite simple groups are determined. In particular, it is proved that if G is a finite simple non-Abelian group, and N is some of its nilpotent subgroups, then |N|²<|G|. Supported through FP “Integration” project No. 274, by RFFR grant No. 99-01-00550, by International Soros Education Program for Exact Sciences (ISEP) grant No. S99-56, and by a SO RAN grant for Young Scientists, Presidium Decree No. 83 of 03/10/2000. Translated fromAlgebra i Logika, Vol. 39, No. 5, pp. 526–546, September—October, 2000.