Model theory of groups and automorphism groups : Blaubeuren, August 1995 / edited by David M. Evans.

Format:

Book

Contributor:

Evans, David M. (David Mark), 1964- editor.

Series:

London Mathematical Society lecture note series ; 244.

London Mathematical Society lecture note series ; 244

Language:

English

Subjects (All):

Model theory.

Group theory.

Physical Description:

1 online resource (xvi, 212 pages) : digital, PDF file(s).

Other Title:

Model Theory of Groups & Automorphism Groups

Place of Publication:

Cambridge : Cambridge University Press, 1997.

Language Note:

English

Summary:

This volume surveys recent interactions between model theory and other branches of mathematics, notably group theory. Beginning with an introductory chapter describing relevant background material, the book contains contributions from many leading international figures in this area. Topics described include automorphism groups of algebraically closed fields, the model theory of pseudo-finite fields and applications to the subgroup structure of finite Chevalley groups. Model theory of modules, and aspects of model theory of various classes of groups, including free groups, are also discussed. The book also contains the first comprehensive survey of finite covers. Many new proofs and simplifications of recent results are presented and the articles contain many open problems. This book will be a suitable guide for graduate students and a useful reference for researchers working in model theory and algebra.

Contents:

Cover; Title; Copyright; Contents; Introduction; Finite covers; 0 Outline of the Notes; 1 Introduction to covers; 1.1 Definitions; 1.2 Examples; 1.3 Related Notions; 1.4 Topological arguments; 1.5 Kernels; 1.6 The model-theoretic context; 1.7 An overview; 2 General constructions; 2.1 Free covers; 2.2 Digraph coverings; 2.3 Coverings of two-graphs; 3 Reductions and special classes of covers; 3.1 Split covers; 3.2 Regular covers and simple covers; 3.3 Minimal covers; 3.4 Irreducibility conditions; 3.5 Superlinked covers; 4 Finite covers with finite kernels; 4.1 Elementary reductions

4.2 Graphic triples and digraphs4.3 Strong types; 4.4 A vector space covering its projective space; 5 The cover problem and independence; 5.1 Strongly determined types; 5.2 Universal covers; 5.3 Highly homogeneous structures; 6 Symmetric extensions with abelian kernels; 6.1 A strategy; 6.2 Symmetric expansions of symmetric extensions; 6.3 Applications of Pontriagin duality; 6.4 Derivations and H1c; 6.5 Finite covers of V(N0,2) and [D]k; 6.6 Cohomology and two-graphs; 7 Computing cohomology groups; 7.1 Dimension shifting and Shapiro's lemma; 7.2 Finiteness results; 8 Problems

Definable subgroups of algebraic groups over pseudo-finite fieldsNotations and conventions; 1. Preliminaries in algebraic geometry: definitions and main facts; 2. Preliminaries on finite and pseudo-finite fields; 3. Definable subgroups of algebraic groups over finite and pseudo-finitefields; 4. Definability of maximal subgroups; References; Groups in pseudofinite fields; 1 An example: one-dimensional matrix groups; 2 Algebraic groups; 3 Irreducible sets and connected groups; 4 Dimension in pseudofinite fields; 5 Affine groups in pseudofinite fields; 6 Almost simple groups

7 Reduction at a prime8 A theorem on reduction; References; The group of automorphisms of the field of complex numbers leaving fixed the algebraic numbers is simple; References; The automorphism group of the field of complex numbers is complete; 1 Introduction and notation; 2 Proof of Proposition 5; 3 Proof of Theorem 1; 4 Generalisations; References; The algebra of an age; 1 The algebra; 2 Polynomial algebras; 3 Transitive extensions; References; Elimination of inverses in groups; Model-theoretic properties of polycyclic-by-finite groups; References; Non-standard free groups; References

4 Scalars in other sorts and Endomorphism Rings of Localised Functors

Notes:

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Includes bibliographical references.

ISBN:

1-139-88234-1

1-107-36744-1

1-107-37201-1

1-107-36253-9

1-107-36820-0

1-299-40507-X

1-107-36498-1

0-511-62917-6