Topics in the model theory of abelian and nilpotent groups

Abstract

This thesis falls naturally into two parts, each concerned with the model theory of a different variety of groups. The algebraist will find, in the preliminary chapter, a survey of the necessary model theory.

A classification of abelian groups by their stability properties has been obtained from results of Eklof and Fisher on saturated abelian groups using Shelah's theorem relating saturation and stability. In Chapter two we develop a direct approach to this problem. We obtain a simple formula for calculating the exact cardinality of the Stone space of a given abelian group. We are then able to distinguish between the various stable classes giving new necessary and sufficient conditions for an abelian group to be superstable. Our method generalises easily to modules over Dedekind domains.

The third chapter contains answers to the question of how much saturation or stability is preserved by the free product, in the variety of all nil-2 groups. First we consider the nil-2 free product of groups, one factor being finite. The elements of such a product are shown to possess a unique normal form which we use to prove, under certain conditions, a "Feferman-Vaught style" theorem for *. As a consequence we obtain a condition sufficient for * to preserve both saturation and stability. In the case of saturation, this condition is shown also to be necessary. These results are extended to products of bounded nil-2 groups, the key being a restricted distributive law for * over the direct product. Finally, we classify numerous nil-2 groups by their stability properties, Some questions are left open, the most interestingof which is whether * preserves o-stability. Results on the absolutely free product of groups, showing it preserves neither model-theoretic property, are also included in this chapter.