Stable groups / Bruno Poizat.

Format:

Book

Author/Creator:

Poizat, Bruno, author.

Series:

Mathematical surveys and monographs ; no. 87.

Mathematical surveys and monographs, 0076-5376 ; volume 87

Standardized Title:

Groupes stables. English

Language:

English

French

Subjects (All):

Model theory.

Group theory.

Physical Description:

1 online resource (145 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2001]

Language Note:

French

Summary:

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.

Contents:

""Contents""; ""A foreword to the English edition""; ""A couple of words about groups""; ""Introduction""; ""Chapter 1. Chain. The chain condition for stable groups""; ""1.1. Stable groups; examples""; ""1.2. The order property""; ""1.3. Chain conditions on subgroups""; ""1.4. Connected components""; ""1.5. Omega-categorical stable groups""; ""1.6. Historic and bibliographic notes""; ""Chapter 2. Structure. Model-theoretic analysis of groups of finite Morley rank""; ""2.1. Generics""; ""2.2. Indecomposable sets; Zil'ber's Theorem""; ""2.3. Lascar analysis of a group of finite Morley rank""

""2.4. Borovik groups and groups of finite Morley rank""""2.5. Binding groups""; ""2.6. Hrushovski analysis""; ""2.7. Historic and bibliographic notes""; ""Chapter 3. Fields. Algebraic properties of groups of finite Morley rank""; ""3.1. Fields of finite Morley rank""; ""3.2. Action of an abelian group on an abelian group""; ""3.3. Minimal groups""; ""3.4. Commutators""; ""3.5. Solvable groups""; ""3.6. Semisimple groups""; ""3.7. Action of a group on a strongly minimal set""; ""3.8. Bad groups""; ""3.9. Historic and bibliographic notes""

""Chapter 4. Geometry. Introduction to algebraic groups""""4.1. Constructible groups""; ""4.2. The Zariski closed sets""; ""4.3. Morphisms and constructible functions""; ""4.4. Varieties""; ""4.5. Algebraic groups; Weil-Hrushovski theorem""; ""4.6. Linear groups; Rosenlicht's theorem""; ""4.7. The Borel-Tits theorem""; ""4.8. Historic and bibliographic notes""; ""Chapter 5. Generics. The reconstruction of a stable group from its generic fragments""; ""5.1. Generic types""; ""5.2. The stratified order""; ""5.3. The superstable case""; ""5.4. Infinitely definable groups""

""5.5. Unidimensional theories""""5.6. Groups given generically""; ""5.7. Historic and bibliographic notes""; ""Chapter 6. Rank. Superstable groups""; ""6.1. Lascar inequalities""; ""6.2. Berline-Lascar Decomposition""; ""6.3. Superstable fields""; ""6.4. Historic and bibliographic notes""; ""Chapter 7. Weight. The last trick for shining in the salons""; ""7.1. Groups and fields with regular generic types""; ""7.2. Hrushovski analysis""; ""7.3. Internal groups of small weight""; ""7.4. Historic and bibliographic notes""; ""Bibliography""; ""Index""; ""A""; ""B""; ""C""; ""D""; ""E""; ""F""

""G""""H""; ""I""; ""L""; ""M""; ""N""; ""O""; ""P""; ""Q""; ""R""; ""S""; ""T""; ""U""; ""V""; ""W""; ""Z""; ""Postscript: Thirteen years later""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 123-124) and index.

Description based on print version record.

ISBN:

1-4704-1314-0