0 - Groups & Goals

Summary

Introduction

The study of stable groups began with the classification of uncountably categorical abelian groups [69] and fields [70] by Angus Macintyre. In 1965 Michael Morley [75] had proved his Categoricity Theorem and defined wstability on the way, and Saharon Shelah [102, 103] embarked on his study of the models of complete first-order theories using his new notion of “stability”; these logical constraints were now being applied to algebraic structures. Differentially closed fields were studied by Shelah in [104], minimal groups by Joachim Reineke in [99], and finally simple groups by Gregory Cherlin in [34], where he formulated his famous conjecture: a simple group of finite Morley rank is an algebraic group over an algebraically closed field. Like Boris Zil'ber's conjecture from [134], of which it may be considered an analogue for groups, it is more programme than claim: one should try to do algebraic geometry, and in particular to define the Zariski topology, by model-theoretic means. Although the original conjecture was refuted by a counter-example of Ehud Hrushovski [57], the programme itself was completed successfully in [59] and has already led to the first applications outside of logic in Hrushovski's proof of the relative version of the Mordell-Lang Conjecture [56].

Meanwhile people had started to look at more general stability classes: Cherlin and Shelah were studying superstable division rings [33], fields and groups [37], Chantal Berline and Daniel Lascar [16, 17, 18] generalized many results from stable to superstable groups, and John Baldwin and Jan Saxl analysed stable groups in [4].