On trifactorized soluble groups of finite rank

Let the soluble-by-finite group G=AB=AC=BC be the product of two nilpotent subgroups A and B and a subgroup C. It is shown that, if G has finite abelian section rank and C is hypercentral (hypercyclic), then G is hypercentral (hypercyclic). Moreover, if G is an L ₁-group and C is nilpotent, then G is nilpotent.Dedicated to Professor Guido Zappa on the occasion of his 75th birthday     

On soluble groups of module automorphisms of finite rank

Let R be a commutative ring, M an R-module and G a group of R-automorphisms of M, usually with some sort of rank restriction on G. We study the transfer of hypotheses between M/C _M (G) and [M,G] such as Noetherian or having finite composition length. In this we extend recent work of Dixon, Kurdachenko and Otal and of Kurdachenko, Subbotin and Chupordia. For example, suppose [M,G] is R-Noetherian. If G has finite rank, then M/C _M (G) also is R-Noetherian. Further, if [M,G] is R-Noetherian and if only certain abelian sections of G have finite rank, then G has finite rank and is soluble-by-finite. If M/C _M (G) is R-Noetherian and G has finite rank, then [M,G] need not be R-Noetherian.     

Approximability of finite rank soluble groups by certain classes of finite groups

For soluble groups of finite rank we obtain the necessary and sufficient condition to be a virtually residually finite p-group. We also prove that a soluble group G of finite rank is residually π-finite for some finite set π of primes if and only if it has no subgroups of type Q and the torsion radical of G is a finite group.     

Normal series of soluble groups of finite rank

Archiv der Mathematik -     

Some soluble groups of finite rank and some related matrix groups

Let G be a torsion-free soluble group of finite rank and F any field. The group algebra FG is an Ore domain; let D denote its division ring of quotients. It seems likely that D is always locally residually finite-dimensional over F. This is certainly so in the non-modular case. Here in some special situations we settle the modular case. We include some applications to groups of matrices.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 894–901, July–August, 1991.     

On residually finite groups of finite general rank

Following A. I.Mal’tsev, we say that a group G has finite general rank if there is a positive integer r such that every finite set of elements of G is contained in some r-generated subgroup. Several known theorems concerning finitely generated residually finite groups are generalized here to the case of residually finite groups of finite general rank. For example, it is proved that the families of all finite homomorphic images of a residually finite group of finite general rank and of the quotient of the group by a nonidentity normal subgroup are different. Special cases of this result are a similar result of Moldavanskii on finitely generated residually finite groups and the following assertion: every residually finite group of finite general rank is Hopfian. This assertion generalizes a similarMal’tsev result on the Hopf property of every finitely generated residually finite group.     

On finite products of soluble groups

Let the finite groupG =AB be the product of two soluble subgroupsA andB, and letπ be a set of primes. We investigate under which conditions for the maximal normalπ-subgroups ofA, B andG the following holds:O _π (G) ∩O _π (G) ⊆O _π (G). The second author would like to thank the Department of Mathematics of the University of Mainz and the Mathematische Forschungsinstitut Oberwolfach for their excellent hospitality during the preparation of this paper.     

On Weyl groups in minimal simple groups of finite Morley rank

We prove that generous non-nilpotent Borel subgroups of connected minimal simple groups of finite Morley rank are self-normalizing. We use this to introduce a uniform approach to the analysis of connected minimal simple groups of finite Morley rank through a case division incorporating four mutually exclusive classes of groups. We use these to analyze Carter subgroups and Weyl groups in connected minimal simple groups of finite Morley rank. Finally, the self-normalization theorem is applied to give a new proof of an important step in the classification of simple groups of finite Morley rank of odd type.     

On the cohomology of finite soluble groups

Cohomological vanishing theorems in dimensions 1 and 2 are established for finite soluble groups. Applications are given to splitting and conjugacy theorems, and also to the structure of automorphism groups of finite soluble groups with nilpotent length at most 3.     

On subgroups of finite soluble groups III

We define a notion of generalized normality for subgroups of finite soluble groups. We show that a normalizer relative to this notion exists and is homomorphism-invariant. We make comparisons with previous constructions, and develop briefly a general theory of normality relations and normalizers.     

On subgroup functors of finite soluble groups

The principal aim of this paper is to study the regular and transitive subgroup functors in the universe of all finite soluble groups. We prove that they form a complemented and non-modular lattice containing two relevant sublattices. This is the answer to a question (Question 1.2.12) proposed by Skiba (1997) in the finite soluble universe.