Lecture Notes in Mathematics. Editors: A. Dold, Heidelberg F. Takens, Groningen B. Tcissier, Paris

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1 Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen B. Tcissier, Paris 1698

2 Springer Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Singapore Tokyo

3 M. Bhattacharjee D. Macpherson R. G. Moller P. M. Neumann Notes on Infinite Permutation Groups Springer

4 Authors Meenaxi Bhattacharjee Department of Mathematics Indian Institute of Technology Guwahati Pan bazar, Guwahati I, India Dugald Macpherson Department of Pure Mathematics University of Leeds Leeds LS2 9JT, UK Rognvaldur G. M()ller Science Institute University of Iceland IS-I07 Reykjavik, Iceland Peter M. Neumann The Queen's College Oxford OXI 4AW, UK Copyright 1997 Hindustan Book Agency (India) and Copyright 1998 Springer-Verlag Berlin Heidelberg Published in cooperation with Hindustan Book Agency (India), New Delhi Outside India exclusive distribution rights remain with Springer-Verlag Berlin Heidelberg Notes of p. 1698> Library of. Congress Cataloging-in-Publication Data infinite permutatlon groups I M. Bhattachar Jee... let al.]. cm. -- (Lecture notes ln mathematlcs, ISSN ; Includes b t b l iographical references and index. ISBN (softcover) 1. Permutation groups. I. BhattacharJee, M. (Meenaxi). II. Series: Lecture notes in mathematics (SprInger-Verlag) QA3.L28 no [QAI75] 510 s--dc21 [512.2] Mathematics Subject Classification (1991): 20B07, 20B 15, 03C ; CIP ISSN ISBN Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved. whether the whole or part of the material is concerned, specifically the rights of translation. reprinting, re-use of illustrations. recitation. broadcasting. reproduction on microfilms or in any other way. and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September in its current version. and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg 1998 Printed in Germany The usc of general descriptive names. registered names, trademarks. etc. in this publication does not imply. even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: / Printed on acid-free paper

Preface The oldest part of group theory is that which deals with finite groups of permutations. One of the newest is the theory of infinite permutation groups.

5 Preface The oldest part of group theory is that which deals with finite groups of permutations. One of the newest is the theory of infinite permutation groups. Much progress has been made during the last two decades, but much remains to be discovered. It is therefore an excellent research area: on the one hand there is plenty to be done; on the other hand techniques are becoming available. Some research takes as its goal the generalisation, extension or adaptation of classical results about finite permutation groups to the infinite case. But mostly the problems of interest in the finite and infinite contexts are quite different. In spite of its newness, the latter has already developed a momentum and an ethos of its own. It has also developed strong links with logic, especially with model theory. To bring this lively and exciting subject to the attention of mathematicians in Assam both students and established scholars a course of sixteen lectures was presented in August and September 1996 at the Indian Institute of Technology, Guwahati, In the available time it was not possible to explore more than a fraction of the area. The lectures were therefore conceived with restricted aims: firstly, to expound a useful amount of general theory; secondly, to introduce and survey just one of the rich areas of recent research in which considerable progress has been made, namely, the theory of Jordan groups. That limited aim is reflected in this book. Although it is a considerable expansion of the lecture notes (and we have included many exercises, which range in difficulty from routine juggling of definitions to substantial recent research results), it is not intended to introduce the reader to more than a small though important and interesting part of the subject. A course at this level can only succeed if it is a collaboration between lecturers and audience. That was so in this case and although the four speakers are those listed as authors of this volume, credit would be spread much more widely if title page conventions permit

6 VI Preface ted. In particular, we record our very warm thanks to the audience for being so enthusiastic and alert, and especially to the three note-takers Dr. B. K. Sharma, Ms. Shreemayee Bora and Ms. Shabeena Ahmed. We are very grateful to Dr. S. Ponnusarny and Dr. K. S. Venkatesh for help with typesetting and with the figures. We wish to thank Prof. Moloy Dutta for help with typing and editing, and, Shabeena and Shreemayee, a second time, for help with proof-reading. We also thank the Editors of the TRIM series, specially Professors R. Bhatia and C. Musili, for accepting the book for publication in the series. MB (Guwahati), HDM (Leeds), RGM (Reykjavik). I1MN (Oxford): April 1997 Some more acknowledgements: The three of us who came from overseas have warm thanks to record to various institutions that provided support: to the Royal Society of London (Macpherson and Neumann); to the Indian National Science Academy (Neumann); to the Indian Institute of Technology Guwahati (Macpherson, Moller and Neumann) for a very kind welcome, for academic facilities and for help with travel within India. Our warmest thanks and congratulations, however, are reserved for our colleague and co-author Dr Meenaxi Bhattacharjee. She conceived and organised the whole project, she provided delightful company and hospitality in Assam, and, with characteristic energy she has brought this book project to a satisfactory conclusion. HDM (Leeds), RGM (Reykjavik), limn (Oxford): April 1997

7 About the lecture course: This book is based on lectures delivered by the four authors at the 'Lecture Course on Infinite Permutation Groups' that was held at the Indian Institute of Technology (IIT) Guwahati, India from 6th August to 19th September, The course was originally conceived as an informal series of lectures coinciding with the visits of Dr. Neumann, Dr. Macpherson and Dr. Moller to the Institute during that period. The lectures were aimed at graduate students, research scholars, teachers and research workers of mathematics drawn from the colleges, universities, research institutes and other academic institutions in this region. The lectures (each lasting 90 minutes) were held twice a week. The course generated a tremendous amount of interest and as many as 60 participants attended the lectures. The first speaker was Dr. Moller who introduced the participants to the basics of permutation group theory. I spoke next on wreath products of groups. In his lectures, Dr. Neumann explained the general theory of Jordan groups. Dr. Macpherson delivered the concluding lectures in which he developed the theory further, leading to the classification of infinite primitive Jordan groups. He also introduced the audience to the basic concepts of model theory, and illustrated the connections between model theory and Jordan groups using the Hrushovski construction. The sixteen chapters of this book correspond roughly to the sixteen lectures delivered at the course. We wish to record our thanks to Professors M. S. Raghunathan and S. G. Dani of the National Board of Higher Mathematics for extending financial support, and to the faculty and staff of IIT Guwahati for their help in organising the course. We are particularly grateful to the Director of IIT Guwahati, Prof. D. N. Buragohain, for providing all infra structural facilities and for his kind encouragement and support. We are also grateful to the Head of the Department of Mathematics at IIT Guwahati, Prof. P. Bhattacharyya, for giving Vll

8 viii Preface us his whole-hearted help and guidance. On a personal note. my son Amlan deserves a special word of thanks for patiently tolerating my total preoccupation with the course last summer and then with the preparation of this book over the past few months. I extend my warm and sincere thanks to each of my three coauthors-my teacher and gut'u Dr. Peter M. Neumann. my friend and collaborator Dr. Dugald Macpherson, and, my friend and qurubluii Dr. Rognvaldur G. Moiler-e-on my own behalf as well as on behalf of everyone else involved in the project, for accepting our invitation to visit Guwahati (and smilingly putting up with many odds during their Indian sojourn), for agreeing to speak at the course, for delivering such superb lectures and for being readily available to the participants for consultation at other times. By doing so they have given a rare opportunity to the participants to benefit from their expertise in the subject. Co-ordinating the course was both thrilling as well as challenging-and I have gained a lot from the experience. MB (Guwahati): April 1997 Addresses of the authors: Dr. Meenaxi Bhattacharjee Department of Mathematics, Indian Institute of Technology, Guwahati Panbazar, Guwahati India meenaxi@iitg.ernet.in Dr. Rognvaldur G. Moller Science Institute. University of Iceland Reykjavik Iceland roggh1jraunvis.hi.is Dr. Dugald Macpherson Department of Pure Mathematics, University of Leeds Leeds LS2 9JT; England pmthdm@amsta.leeds.ac.uk Dr. Peter M. Neumann The Queen's College Oxford OX1 4AW England Peter.Neumanuwqueens.ox.ac.uk

9 Contents 1 Some Group Theory 1 2 Groups acting on Sets Group actions Permutation groups Cycles and cycle types Basic facts about symmetric groups 16 3 Transitivity Orbits and transitivity Stabilisers and transitivity Extensions of transitivity Homogeneity Primitivity G-congruences Primitive spaces Extensions of primitivity Suborbits and Orbitals Suborbits and orbitals Orbital graphs and primitivity 41 6 More about Symmetric Groups Normal subgroups of symmetric groups Groups with finitary permutations 52 7 Linear Groups 7.1 General Linear Groups IX

10 x Contents 7.2 Projective groups. 7.3 Affine groups. 7.4 Projective and affine spaces 8 Wreath Products 8.1 Definition Wreath products as permutation groups 8.3 Imprimitivity of wreath products 8.4 Variations of wreath products 9 Rational Numbers 9.1 Cantor's Theorem. 9.2 Back and forth 1'8 going forth. 9.3 Order-automorphisms of the rationals 10 Jordan Groups 10.1 Definitions and some examples 10.2 Basic properties of Jordan sets 10.3 More properties of Jordan sets loa Cofinite Jordan sets Examples of Jordan Groups 11.1 Some examples Finite primitive examples Highly transitive examples Linear groups and Steiner systems 11.3 Linear relational structures Linear order Linear betweenness relation Circular order Separation relation Relations related to betweenness 12.1 Semilinear order C-relations General betweenness relations 12.4 D-relations Primitive groups with primitive Jordan sets n

11 Contents Xl 13 Classification Theorems 13.1 Statement of the theorems Some recognition theorems 13.3 Some basic lemmas Outline of the proof of Theorem Homogeneous Structures 14.1 Some model theory Building homogeneous structures Examples of homogeneous structures 14.4 Proof of Fraisse's Theorem.. 15 The Hrushovski Construction 15.1 Generalisation of Fraisse's Theorem Building a geometry Extending the geometry Applications and Open Questions 16.1 Bounded groups 16.2 Cycle types Strictly minimal sets 'Back and forth' arguments 16.5 Homogeneous structures.. 16.,6 Structure of Jordan groups 16.7 Generalisations of Jordan groups Bibliography List of Symbols Index

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