Eugene Plotkin

Chevalley Groups over Commutative Rings: I. Elementary Calculations

This is the first in a series of papers dedicated to the structure of Chevalley groups over commutative rings. The goal of this series is to systematically develop methods of calculations in Chevalley groups over rings, based on the use of their minimal modules. As an application, we give new direct proofs for normality of the elementary subgroup, description of normal subgroups and similar results due to E. Abe, G. Taddei, L. N. Vaserstein, and others, as well as some generalizations. In this first part we outline the whole project, reproduce construction of Chevalley groups and their elementary subgroups, recall familiar facts about the elementary calculations in these groups, and fix a specific choice of the structure constants.

Automorphisms of categories of free algebras of varieties

Let Θ be an arbitrary variety of algebras and let Θ0 be the category of all free finitely generated algebras from Θ. We study automorphisms of such categories for special Θ. The cases of the varieties of all groups, all semigroups, all modules over a noetherian ring, all associative and commutative algebras over a field are completely investigated. The cases of associative and Lie algebras are also considered. This topic relates to algebraic geometry in arbitrary variety of algebras Θ.

Identities for finite solvable groups and equations in finite simple groups

We characterise the class of finite solvable groups by two-variable identities in a way similar to the characterisation of finite nilpotent groups by Engel identities. Let u1 = x −2 y −1 x, and un+1 =[ xunx −1 ,y uny −1 ]. The main result states that a finite group G is solvable if and only if for some n the identity un(x, y) ≡ 1h olds inG. We also develop a new method to study equations in the Suzuki groups. We believe that, in addition to the main result, the method of proof is of independent interest: it involves surprisingly diverse and deep methods from algebraic and arithmetic geometry, topology, group theory, and computer algebra (Singular and MAGMA).

Net subgroups of Chevalley groups

We introduce and study net subgroups of Chevalley groups of normal and certain twisted types. Another subgroup of Chevalley groups related to a net was discussed in Ref. Zh. Mat. 1976; 10A151; 1977, 10A301; 1978, 6A476.

Visual Basic Representations

We depict the weight diagrams (alias, crystal graphs) of basic and adjoint representations of complex simple Lie algebras/algebraic groups and describe some of their uses.

Two-Variable Identities in Groups and Lie Algebras

We study two-variable Engel-like relations and identities characterizing finite-dimensional solvable Lie algebras and, conjecturally, finite solvable groups and introduce some invariants of finite groups associated with such relations. Bibliography: 29 titles.

Geometrical equivalence of groups

The notion of geometrical equivalence of two algebras, which is basic for this paper, is introduced in [5], [6]. It is motivated in the framework of universal algebraic geometry, in which algebraic varieties are considered in arbitrary varieties of algebras. Universal algebraic geometry (as well as classic algebraic geometry) studies systems of equations and its geometric images, i.e., algebraic varieties, consisting of solutions of equations. Geometrical equivalence of algebras means, in some sense, equal possibilities for solving systems of equations. In this paper we consider results about geometrical equivalence of algebras, and special attention is paied on groups (abelian and nilpotent).

ALGEBRAIC LOGIC AND LOGICALLY-GEOMETRIC TYPES IN VARIETIES OF ALGEBRAS

The main objective of this paper is to show that the notion of type which was developed within the frames of logic and model theory has deep ties with geometric properties of algebras. These ties go back and forth from universal algebraic geometry to the model theory through the machinery of algebraic logic. We show that types appear naturally as logical kernels in the Galois correspondence between filters in the Halmos algebra of first order formulas with equalities and elementary sets in the corresponding affine space.

A commutator description of the solvable radical of a finite group

We are looking for the smallest integer k> 1providing the following characteri- zation of the solvable radical R.G/ of any finite group G: R.G/ coincides with the collection of all g 2 G such that for any k elements a1 ;a 2 ;:::;a k 2 G the subgroup generated by the elements g; ai ga � 1 i , i D 1; :::;k , is solvable. We consider a similar problem of finding the smallest integer `>1 with the property that R.G/ coincides with the collection of all g 2 G such that for anyelements b1 ;b 2 ;:::;b ` 2 G the subgroup generated by the commutators Œg; bi � , i D 1; :::;` , is solvable. Conjecturally, k DD 3. We prove that both k andare at most 7. In particular, this means that a finite group G is solvable if and only if every 8 conjugate elements of G generate a solvable subgroup.

BURNSIDE-TYPE PROBLEMS RELATED TO SOLVABILITY

In the paper we pose and discuss new Burnside-type problems, where the role of nilpotency is replaced by that of solvability.