a AY RTI TNS Fewest mlseatar eirekee 2 ACN HOS yt SO RE IN TNA FLT\ oP) + 82) LO N ————=—= >aN a —— oO SSISSl 2) —SaS—S= = — —=—=— Digitized by the Internet Archive in 2021 with funding from Kahle/Austin Foundation https://archive.org/details/profinitegroupsO00O0wils LONDON MATHEMATICAL SOCIETY MONOGRAPHS NEW SERIES Series Editors H.G. Dales PeterM. Neumann LONDON MATHEMATICAL SOCIETY MONOGRAPHS NEW SERIES Previous volumes of the LMS Monographs were published by Academic Press, to whom all enquiries should be addressed. Volumes in the New Series will be published by Oxford University Press throughout the world. NEW SERIES 1. Diophantine inequalities R. C. Baker 2. The Schur multiplier Gregory Karpilovsky 3. Existentially closed groups Graham Higman and Elizabeth Scott 4, The asymptotic solution of linear differential systems M.S. P. Eastham 5. The restricted Burnside problem Michael Vaughan-Lee 6. Pluripotential theory Maciej Klimek 7. Free Lie algebras Christophe Reutenauer 8. The restricted Burnside problem (2nd edition) Michael Vaughan-Lee 9. The geometry of topological stability Andrew du Plessis and Terry Wall 10. Spectral decompositions and analytic sheaves J. Eschmeier and M. Putinar 11. Anatlas of Brauer characters C. Jansen, K. Lux, R. Parker, and R. Wilson 12. Fundamentals of semigroup theory John M. Howie 13. Area, lattice points, and exponential sums M.N. Huxley 14. Super-real fields H.G. Dales and W. H. Woodin 15. Integrability, self-duality, and twistor theory L. Mason and N. M. J. Woodhouse 16. Categories of symmetries and infinite-dimensional groups Yu. A. Neretin 17. Interpolation, identification, and sampling Jonathan R. Partington 18. Metric number theory Glyn Harman 19. Profinite groups John S. Wilson Profinite Groups John S. Wilson School of Mathematics and Statistics University of Birmingham CLARENDON PRESS * OXFORD 1998 > A oo Th AY Nr PAA be iL ei F2> ANF y UN4 IV eCxoinv oPlela fk AVEC AALL ASATKTAIC A - FAIRg eBANKS Oxford University Press, Great Clarendon Street, Oxford OX2 6DP Oxford New York Athens Auckland Bangkok Bogota Bombay Buenos Aires Calcutta CapeTown Chennai DaresSalaam Delhi Florence Hong Kong Istanbul Karachi KualaLumpur Madras Madrid Melbourne Mexico City Mumbai Nairobi Paris Sao Paolo Singapore Taipei Tokyo Toronto Warsaw and associated companies in Berlin Ibadan Oxford is a registered trade mark of Oxford University Press Published in the United States by Oxford University Press Inc., New York © John S. Wilson, 1998 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press. Within the UK, exceptions are allowed in respect of any fair dealing for the purpose of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act, 1988, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms and in other countries should be sent to the Rights Department, Oxford University Press, at the address above. This book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, re-sold, hired out, or otherwise circulated without the publisher's prior consent in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser. A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data (Data available ) ISBN O 19 §50082 3 Typeset by the author Printed in Great Britain by Bookcraft (Bath) Ltd Midsomer Norton, Avon Preface Profinite groups are objects of interest for mathematicians working in a variety of areas. For the specialist in abstract groups, they provide the means for focussing attention on properties of finite homomorphic images, and sometimes for deriving strong results having no obvious connection with finite images. For the number theorist, profinite groups are the groups which arise as Galois groups of algebraic field extensions. And for the analyst, they are the quotient groups of compact Hausdorff topological’ groups modulo the connected component of the identity. The theory of profinite groups reflects the diverse sources from which it derives its inspiration: the methods used are for the most part algebraic, but they are leavened with ideas from general topology, and the connections with field theory provide powerful motivation and an interesting perspective. On several occasions between 1982 and 1992, I gave courses of Part III lec- tures on profinite groups for graduate students in Cambridge. They contained enough basic material to enable the audience to tackle a wide variety of prob- lems concerning profinite groups, and proceeded to examine a topic of recent research in some depth. (The topic varied from year to year.) The books which I consulted when preparing the lectures did not present the material on profinite groups at an appropriate level of rigour and sophistication for such a course: they took a very direct route through the subject and skilfully skirted technical obstacles, because their true goals were either in field theory or in the study of profinite groups with rather specific properties. Sections of my lecture notes (complete with mistakes) have been circulating for some years, and in writing this book I am acceding to the requests of a number of colleagues and former students to make my account of the subject more generally available. This book, then, is intended first of all to provide an elementary and rigorous introduction to profinite groups. Its aims are to lay the foundations thoroughly enough for the reader to gain facility in handling profinite groups, to introduce the reader to some aspects of the subject which have received particular atten- tion, and, I hope, to convey some of the qualities which give the study of profinite groups a distinctive character and make it an attractive part of mathematics. It has not been my intention to mention all topics of interest or to reach the frontiers of knowledge, although the latter does happen from time to time, by accident or because of the enthusiasm of the author. Topics from the lectures which seemed too esoteric or difficult have been suppressed. Like the lectures, the book is addressed to graduate students and more experienced mathematicians working in other areas. It is assumed from the beginning that the reader has some familiarity with abstract groups and topological spaces; however the prerequisites from general vi Preface topology are summarized quickly in Chapter 0. In Chapter 3 and various exam- ples later on, a knowledge of classical Galois theory is assumed. Readers who do not have this knowledge could omit all references to Galois theory, but I would urge them strongly instead to redress this gap in their education. All that is needed can be found in any standard text (e.g. Lang’s Algebra). From Chapter 5, a knowledge of linear algebra and some aquaintance with rings and modules are assumed and the pace accelerates gently. Although it is clearly preferable that the reader should work through the whole book from beginning to end, it is possible to find economical routes to some of the highlights and to meet the demands of shorter courses by omitting material in various ways. The foundations are completed in Chapters 0—4; a very approximate indication of dependencies between sections in subsequent chapters is given in the Leitfaden. Because of the requirement of planarity in the Leitfaden we have promoted Pontryagin duality to a position that it deserves but does not quite receive in the text; we have also omitted the obvious dependency of certain sections on simple properties of free pro-p groups. At the end of each chapter there are exercises, followed by bibliographical notes with suggestions for further reading. The exercises should be considered as an integral part of the text and readers are encouraged to do most of them, particularly those in the first few chapters. They range from easy five-finger exercises, through results which might reasonably have been included in the text, to rather challenging problems related to recent research. Many of the exercises are referred to in later chapters. Those exercises which quite clearly assume specialist knowledge, such as familiarity with cardinal arithmetic, can be omitted without prejudicing an understanding of the subject. It is a pleasure to express my indebtedness a number of people who have helped me in the preparation of this book. I myself began to learn the subject from the books of H. Koch (1970) and J.-P. Serre (1964), and the influence of these books cannot be overestimated. I am also grateful to the graduate students and colleagues in Cambridge whose valuable feedback on my lectures was no less influential. I owe a great debt to Professor D. L. Johnson, who read an earlier version of the whole manuscript carefully and drew many infelicities and errors to my attention. Dr P. A. Zalesskii and Mr M. Smith have both read parts of the manuscript and made helpful suggestions. But for the comments of these people, the text would contain many more mistakes than it does, F inally, I owe a debt of a different kind to Dr D. R. H. Jones, for making his college room available to me and so enabling me to complete the preparation of this text as I began it, surrounded by many friends in Christ’s College, Cambridge. Birmingham J.S.W. December 1997