The aim of this paper is to describe an analogue of the theory of nontrivial torsion-free divisible abelian groups for metabelian groups. We obtain illustrations for “old-fashioned” model theoretic algebra and “new” examples in the theory of stable groups. We begin this paper with general considerations about model theory. In the second section we present our results and we give the structure of the rest of the paper. Most parts of this paper use only basic concepts from model theory and (...) group theory. However, in Section 5, we need some somewhat elaborate notions from stability theory. One can find the beginnings of this theory in [14], and we refer the reader to [16] or [21] for stability theory and to [22] for stable groups.§1. Some model theoretic considerations. Denote by the theory of torsion-free abelian groups in the language of groups ℒgp. A finitely generated group G satisfies iff G is isomorphic to a finite direct power of ℤ. It follows that axiomatizes the universal theory of free abelian groups and that the theory of nontrivial torsion-free abelian groups is complete for the universal sentences. Denote by the theory of nontrivial divisible torsion-free abelian groups. (shrink)

In this paper, we shall survey results about the group-theoretic properties of stable groups. These can be classified into three main categories, according to the strength of the assumptions needed: chain conditions, generic types, and some form of rank. Each category has its typical application: Chain conditions often allow us to deduce global properties from local ones, generic properties are used to get definable groups from undefinable ones, and rank is necessary to interpret fields in certain group actions. While originally (...) the main input came from algebraic group theory, founded on the similarities between rank and dimension, increasingly methods borrowed from finite group theory have come to play a rôle, emphasizing the study of characteristic subgroups and characteristic families of subgroups. (shrink)

The elementary theory of a nontrivial free Lie algebra over a commutative integral domain is unstable and has the strict order property.

For every structure M of finite signature Mekler 781) has constructed a group G such that for every κ the maximal number of n -types over an elementary equivalent model of cardinality κ is the same for M and G . These groups are nilpotent of class 2 and of exponent p , where p is a fixed prime greater than 2. We consider stable structures M only and show that M is CM -trivial if and only if G is (...) CM -trivial. Furthermore, we obtain that the free group F 2 in the variety of 2 -nilpotent groups of exponent p>2 with ω free generators has a CM -trivial ω -stable theory. (shrink)

In a recent communication an uncountably categorical group has been constructed that has a non-locally-modular geometry and does not allow the interpretation of a field. We consider a system Δ of elementary axioms fulfilled by some special subgroups of the above group. We show that Δ is complete and stable, but not superstable. It is not even a R-group in the sense discussed by Wagner.