# Stable Groups

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*Bruno Poizat*

This is the English translation of the book originally published in 1987. It is
a faithful reproduction of the original, supplemented by a new Foreword and
brought up to date by a short postscript. The book gives an introduction by a
specialist in contemporary mathematical logic to the model-theoretic study of
groups, i.e., into what can be said about groups, and for that matter, about
all the traditional algebraic objects.

The author introduces the groups of finite Morley rank (those satisfying the
most restrictive assumptions from the point of view of logic), and highlights
their resemblance to algebraic groups, of which they are the prototypes. (All
the necessary prerequisites from algebraic geometry are included in the book.)
Then, whenever possible, generalizations of properties of groups of finite
Morley type to broader classes of superstables and stable groups are
described.

The exposition in the first four chapters can be understood by mathematicians
who have some knowledge of logic (model theory). The last three chapters are
intended for specialists in mathematical logic.

#### Reviews & Endorsements

This is a beautiful book in which almost everything known about stable groups appears.

-- Zentralblatt MATH

#### Table of Contents

# Table of Contents

## Stable Groups

- Contents ix10 free
- A foreword to the English edition xi12 free
- A couple of words about groups 116 free
- Introduction 318
- Chapter 1. Chain. The chain condition for stable groups 1126
- Chapter 2. Structure. Model-theoretic analysis of groups of finite Morley rank 2540
- Chapter 3. Fields. Algebraic properties of groups of finite Morley rank 4560
- 3.1. Fields of finite Morley rank 4560
- 3.2. Action of an abelian group on an abelian group 4863
- 3.3. Minimal groups 5065
- 3.4. Commutators 5368
- 3.5. Solvable groups 5671
- 3.6. Semisimple groups 5873
- 3.7. Action of a group on a strongly minimal set 6075
- 3.8. Bad groups 6378
- 3.9. Historic and bibliographic notes 6883

- Chapter 4. Geometry. Introduction to algebraic groups 6984
- 4.1. Constructible groups 6984
- 4.2. The Zariski closed sets 7186
- 4.3. Morphisms and constructible functions 7590
- 4.4. Varieties 7893
- 4.5. Algebraic groups; Weil-Hrushovski theorem 8196
- 4.6. Linear groups; Rosenlicht's theorem 86101
- 4.7. The Borel-Tits theorem 88103
- 4.8. Historic and bibliographic notes 91106

- Chapter 5. Generics. The reconstruction of a stable group from its generic fragments 93108
- Chapter 6. Rank. Superstable groups 107122
- Chapter 7. Weight. The last trick for shining in the salons 115130
- Bibliography 123138
- Index 125140
- Postscript: Thirteen years later 127142