Oligomorphic permutation groups / Peter J. Cameron.

Includes bibliographical references (pages 145-154) and index.

Print version record.

The study of permutation groups has always been closely associated with that of highly symmetric structures. The objects considered here are countably infinite, but have only finitely many different substructures of any given finite size. They are precisely those structures which are determined by first-order logical axioms together with the assumption of countability. This book concerns such structures, their substructures and their automorphism groups. A wide range of techniques are used: group theory, combinatorics, Baire category and measure among them. The book arose from lectures given at a research symposium and retains their informal style, whilst including as well many recent results from a variety of sources. It concludes with exercises and unsolved research problems.

Cover; Title; Copyright; Preface; Contents; 1 Background; 1.1 History and notation; 1.2 Permutation groups; 1.3 Model theory; 1.4 Category and measure; 1.5 Ramsey's Theorem; 2 Preliminaries; 2.1 The objects of study; 2.2 Reduction to the countable case; 2.3 The canonical relational structure; 2.4 Topology; 2.5 The Ryll-Nardzewski Theorem; 2.6 Homogeneous structures; 2.7 Strong amalgamation; 2.8 Appendix: Two proofs; 2.9 Appendix: Quantifier elimination and model completeness; 2.10 Appendix: The random graph; 3 Examples and growth rates; 3.1 Monotonicity; 3.2 Direct and wreath products

3.3 Some primitive groups3.4 Homogeneity and transitivity; 3.5 fn = fn + 1; 3.6 Growth rates; 3.7 Appendix: Cycle index; 3.8 Appendix: A graded algebra; 4 Subgroups; 4.1 Beginnings; 4.2 A theorem of Macpherson; 4.3 The random graph revisited; 4.4 Measure, continued; 4.5 Category; 4.6 Multicoloured sets; 4.7 Almost all automorphisms?; 4.8 Subgroups of small index; 4.9 Normal subgroups; 4.10 Appendix: The tree of an age; 5. Miscellaneous topics; 5.1 Jordan groups; 5.2 Going forth; 5.3 No-categorical, unstable structures; 5.4 An example; 5.5 Another example; 5.6 Oligomorphic projective groups

5.7 Orbits on infinite setsReferences; Index

eBooks on EBSCOhost EBSCO eBook Subscription Academic Collection - Worldwide