Model Theory of Groups (Notre Dame Mathematical Lectures)

THE MODEL THEORY OF GROUPS

Notre Dame Mathematical Lectures Number 11

THE MODEL THEORY OF GROUPS Ali Nesin and Anand Pillay, editors

UNIVERSITY OF NOTRE DAME PRESS NOTRE DAME, INDIANA

Copyright © 1989 by University of Notre Dame Press Notre Dame, Indiana 46556 All Rights Reserved

Library of Congress Cataloging-in-Publication Data The Model theory of groups. (Notre Dame mathematical lectures; 11 ) 1. Group theory. I. Nesin, AH. II. Pillay, Anand. III. Series: Notre Dame mathematical lectures; no. 11. QA1 .N87 no. 11 a [QA171 ] 510s 88-40322 ISBN 0-268-01371-3 [512'.22] Manufactured in the United States of America

CONTENTS

MODEL THEORY, STABILITY THEORY AND STABLE GROUPS, AnandPillay AN INTRODUCTION TO ALGEBRAICALLY CLOSED FIELDS AND VARIETIES, Bruno Poizat

1

41

COUNTABLY CATEGORICAL EXPANSIONS OF PROJECTTVE SPACES, Simon Thomas 68 GENERIC FORMULAS AND TYPES A LA HODGES, Victor Harnik

88

SOME NOTES ON STABLE GROUPS, John Baldwin

100

NON-ASSOCIATIVE RINGS OF FINITE MORLEY RANK, AliNesin

117

MODULES WITH REGULAR GENERIC TYPES, Ivo Herzog and Philipp Rothmaler

138

MODEL THEORETIC VERSIONS OF WEIL'S THEOREM ON PREGROUPS, Elisabeth Bouscaren

177

ON SUPERSTABLE FIELDS WITH AUTOMORPHISMS, Ehud Hrushovski

186

ON THE EXISTENCE OF 0-DEFINABLE NORMAL SUBGROUPS OF A STABLE GROUP,

AnandPillay

THE GROUP CONFIGURATION - AFTER E. HRUSHOVSKI, Elisabeth Bouscaren v

192

199

Preface This volume contains contributions by participants in the Stable Group seminar at Notre Dame. This seminar began during the 1985-86 Notre Dame Logic Year and continued up to December 1987. Except for Simon Thomas' paper on expansions of projective spaces, all the papers here are connected in one way or another with stable group theory. This volume contains introductory, expository and research papers in the area and we hope will be of interest to beginners as well as "experts". We cover much of the material in Poizat's book Groupes Stables and go beyond it in some respects (for example Bouscaren's paper on Hrushovski's important group configuration theorem in which the presence of a definable group is recognized from a certain forking configuration). So we hope our volume will serve as a useful complement to Poizat's book. We would like to thank the University of Notre Dame (Mathematics Department and College of Science), the National Science Foundation (Grant DMS 85 09920) and the Argonne Universities Association Trust Fund, for their support of the 1985-86 Logic Year at Notre Dame, which in addition to funding a conference in April 1986, allowed us to invite many of the contributors to this volume. We would also like to thank Joan Hoerstman, Karen Jacobs and Leona Strickland for their excellent and patient work in preparing the manuscript. AliNesin Anand Pillay

Notre Dame, Indiana October 1988

MODEL THEORY, STABILITY THEORY & STABLE GROUPS Anand Pillay* The aim of this chapter is to introduce the reader to the theory of stable groups not to give a rigorous exposition of the general theory. Thus we tend to proceed from the concrete to the abstract, with several examples and analyses of special cases along the way. On the other hand, getting to grips with stable groups presupposes some understanding of the point of view of model theory in general and stability theory in particular, and the first few sections are devoted to the latter. 1. MODEL THEORY By a relational structure M we understand a set M (called the universe or underlying set of M) equipped with relations Rj of arity ni < co say, for n

lei.

i

Namely, for i e I, R± is a subset of the Cartesian product M .

Here I and depend on M and are called the signature of M. We also insist that I always contains a distinguished element i= such that Ri- is the diagonal {(a,a) : ae M} c, M2. Often the distinction between M and M is blurred. The model theorist is interested in certain subsets of M and of Mn (the definable sets) which are obtained in a simple fashion from the RI. So J9(M) is a collection of subsets of Mn, n< co, which can be characterized as follows: (i) Every RIE J9(M). (ii) If n < co, Xe J9(M) is a subset of Mn and n is a permutation of {l,...,n} then 7c(X) = {(a^i) .....a^n)) : (ai,...,an) e X} e £)(M). (iii) J9(M) is closed under Boolean combinations, i.e. if n < co and

* Partially supported by NSF Grant DMS 8601289

2

Model Theory, Stability Theory & Stable Groups

X,Y e JB(M) are subsets of M" then XU Y , X f l Y , M n - X areallin JB(M). (iv) If Xe JD(M) andYe J9(M) then Xx Ye J9(M). (v) If X e J0(M) is a subset of Mn+m, then the projection of X on n M is in J9(M). (vi) If Xe J9(M) is a subset of Mn+m and âe Mn then Xa = {be Mm: (a,b) e X} is in J9(M). (vii) Nothing else is in J9(M). We call J9(M) the class of definable sets of M. These definable sets can be defined (and usually are) syntactically. Associated to the relational structure M (in fact to its signature) is a language L(M) consisting of symbols: Pi for each ie I, "variables" Xj for each j-,ao(n)) e p where a =7C"1If \\f is (3xn+i) 9 and xn+i is a free variable of 9, then \|/(xi,...,xn) is true of (ai,...,an) in M if there is an+i e M such that 9(xi,...,xn,xn+i) is true of (ai,...,an+i) in M. Similarly for y = (Vxn+i) 9. The clauses for A,v, -i are obvious. We abbreviate "9(xi,...,xn) is true of (ai,...,an) in M" by the notation M 1= 9(ai,...,an). (Note this notation depends on our having listed the free variables in 9 in a certain order). By abuse of everything, we can and will think of M N= 9(ai,...,an) as saying that 9 is true when we substitute ai for xj.

AnandPillay

3

It is now routine to check that Fact 1.1. If X c Mn, then X e J9(M) if and only if there are an L(M) formula