Automorphism groups of nilpotent groups

We construct a full class of nilpotent groups of class 2 of an arbitrary infinite cardinality . Their centers, commutator subgroups and factors modulo the center will be the same and a homogeneous direct sum of a group of rank 1 or 2. Their automorphism groups will coincide and the factor group modulo the stabilizer could be an arbitrary group of size $\leqq$ .     

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.     

On universal nilpotent groups

AbstractWe show that some classes of nilpotent groups have countable universal members and others do not.Presented by Bjarni Jónsson.     

Torsion-free nilpotent groups andE-modules

Research partially supported by NSF Grant no. DMS 8701074, and Baylor University's Summer Sabbatical Program.     

On A-Critical nilpotent groups

It is useful to consider a critical structure of a class of groups. For example, the results on minimal non-nilpotent group, and on minimal non-p-nilpotent group, have been widely used in group theory. In this paper, we analyze the critical structure of a class of groups, which admit an operator group.     

Semivarieties of nilpotent groups

Semivarieties of groups are quasivarieties defined by quasi-identities of the form t = 1 → f = 1. It is proved that a set of semivarieties in every variety of class two nilpotent p-groups of finite exponent having a commutator subgroup of exponent p (p is a prime) is at most countable. It is stated that a variety of class two nilpotent groups with commutator subgroup of exponent p contains a set of semivarieties of the cardinality of the continuum.     

Existentially complete nilpotent groups

Letn be a positive integer, letK _n denote the theory of groups nilpotent of class at mostn, and letK _n ⁺ denote the theory of torsion-free groups nilpotent of class at mostn. We show that ifn≧2 then neitherK _n norK _n ⁺ has a model companion. ForK _n we obtain the stronger result that the class of finitely generic models is disjoint from the class of infinitely generic models. We also give some other results about existentially complete nilpotent groups. Dedicated to the Memory of Abraham Robinson.     

Isoperimetry of nilpotent groups

This survey gives an overview of the isoperimetric properties of nilpotent groups and Lie groups. It discusses results for Dehn functions and filling functions as well as the techniques used to retrieve them. The content reaches from long standing results up to the most recent development.     

On constructive nilpotent groups

We prove the following: (1) a torsion-free class 2 nilpotent group is constructivizable if and only if it is isomorphic to the extension of some constructive abelian group included in the center of the group by some constructive torsion-free abelian group and some recursive system of factors; (2) a constructivizable torsion-free class 2 nilpotent group whose commutant has finite rank is orderably constructivizable.     

Automorphism groups of free metabelian nilpotent groups

Published in Algebra i Logika, Vol. 29, No. 6, pp. 746–751, November–December, 1990.     

Automorphism groups of nilpotent groups and spaces

A pair of finitely generated, torsion-free nilpotent groups G₁,G₂ is constructed with the properties that G₁ and G₂ are p-isomorphic for all primes p, yet Aut(G₁) and Aut(G₂) are not isomorphic. The example constructed is compared to an analogous example in the homotopy category of simply connected, finite CW-complexes.