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The Automorphism Tower Problem Simon Thomas The Automorphism Tower Problem Simon Thomas Rutgers University, New Brunswick, New Jersey 08903 E-mail address: [email protected] Contents Preface vii Introduction ix Chapter 1. The Automorphism Tower Problem 1 1.1. Automorphism towers 1 1.2. Some fundamental results 6 1.3. Some examples of automorphism towers 10 1.4. The infinite dihedral group 19 1.5. Notes 23 Chapter 2. Wielandt's Theorem 25 2.1. Automorphism towers of finite groups 25 2.2. Subnormal subgroups 31 2.3. Finite p-groups 35 2.4. Notes 39 Chapter 3. The Automorphism Tower Theorem 41 3.1. The automorphism tower theorem 41 3.2. τκ is increasing 47 3.3. A better bound 51 3.4. The automorphism tower problem revisited 57 3.5. Notes 60 Chapter 4. The Normaliser Tower Technique 61 4.1. Normaliser towers 62 4.2. Coding structures in graphs 68 4.3. Coding graphs in fields 71 4.4. A technical lemma 78 iii iv CONTENTS 4.5. Notes 88 Chapter 5. Hamkins' Theorem 89 5.1. Automorphism towers of arbitrary groups 89 5.2. Two examples and many questions 94 5.3. Notes 98 Chapter 6. Set-theoretic Forcing 101 6.1. Countable transitive models of ZFC 102 6.2. Set-theoretic forcing 108 6.3. Preserving cardinals 116 6.4. Nice P-names 125 6.5. Some observations and conventions 127 6.6. τ(G) is not absolute 131 6.7. An absoluteness theorem for automorphism towers 139 6.8. Iterated forcing 141 6.9. Notes 149 Chapter 7. Forcing Long Automorphism Towers 151 7.1. The nonexistence of a better upper bound 152 7.2. Realising normaliser towers within infinite symmetric groups 158 7.3. Closed groups of uncountable degree 164 7.4. τκ can be strictly increasing 170 7.5. Two more applications 172 7.6. The main gap 176 7.7. Notes 177 Chapter 8. Changing The Heights Of Automorphism Towers 179 8.1. Changing the heights of automorphism towers 180 8.2. More normaliser towers 186 8.3. Rigid trees 196 8.4. Notes 209 Chapter 9. Bounding The Heights Of Automorphism Towers 211 9.1. The overspill argument 211 CONTENTS v 9.2. Conservative extensions 217 9.3. The reflection argument 221 9.4. Notes 230 Index 231 Bibliography 235 Preface This is a preliminary version of an unpublished book on the automorphism tower problem, which was intended to be intelligible to beginning graduate students in both logic and algebra. Simon Thomas vii Introduction If G is a centreless group, then there is a natural embedding of G into its automorphism group Aut(G), obtained by sending each g 2 G to the corresponding inner automorphism ig 2 Aut(G). It is easily shown that the group Inn(G) of inner automorphisms is a normal subgroup of Aut(G) and that CAut(G)(Inn(G)) = 1. In particular, Aut(G) is also a centreless group. This enables us to define the automorphism tower of G to be the ascending chain of centreless groups G = G0 E G1 E G2 E :::Gα E Gα+1 E ::: such that for each ordinal α (a) Gα+1 = Aut(Gα); and S (b) if α is a limit ordinal, then Gα = Gβ. β<α (At each successor step, we identify Gα with Inn(Gα) via the natural embedding.) The automorphism tower is said to terminate if there exists an ordinal α such that Gα+1 = Gα. Of course, this occurs if and only if there exists an ordinal α such that Aut(Gα) = Inn(Gα). In this case, the height τ(G) of the automorphism tower is defined to be the least ordinal α such that Gα+1 = Gα. In 1939, Wielandt proved that if G is a finite centreless group, then its automorphism tower terminates after finitely many steps. However, there exist natural examples of infinite centreless groups whose automorphism towers do not terminate in finitely many steps. For example, it can be shown that the automorphism tower of the infinite dihedral group D1 terminates after exactly ! +1 steps. The classical version of the automorphism tower problem asks whether the automorphism tower of an arbitrary centreless group G eventually terminates, perhaps after a transfinite number of steps. In the 1970s, a number of special cases of the automorphism tower problem were solved. For example, Rae and Roseblade proved that the automorphism tower of a centreless Cernikovˇ group terminates after finitely many steps and Hulse proved ix x INTRODUCTION that the automorphism tower of a centreless polycyclic group terminates after countably many steps. In each of these special cases, the proof depended upon a detailed understanding of the groups Gα occurring in the automorphism towers of the relevant groups G. But the problem was not solved in full generality until 1984, when I proved that the automorphism tower of an arbitrary centreless group + G terminates after at most 2jGj steps. The proof is extremely simple and uses only the most basic results on automorphism towers, together with some elemen- tary properties of the infinite cardinal numbers. Of course, this still leaves open the problem of determining the best possible upper bound for the height τ(G) of the automorphism tower of an infinite centreless group G. For example, it is natural to ask whether there exists a fixed cardinal κ such that τ(G) ≤ κ for every infinite centreless group G. This question turns out to be relatively straightforward. For each ordinal α, it is possible to construct an infinite centreless group G such that τ(G) = α. However, if we ask whether it is true that τ(G) ≤ 2jGj for every infinite centreless group G, then matters become much more interesting. For example, it is independent of the classical ZFC axioms of set theory whether τ(G) ≤ 2@1 for every centreless group G of cardinality @1; i.e. this statement can neither be proved nor disproved using ZFC. This book presents a self-contained account of the automorphism tower prob- lem, which is intended to be intelligible to beginning graduate students in both logic and algebra. There are essentially no set-theoretic prerequisites. The only requirement is a basic familiarity with some of the fundamental notions of algebra. The first half of the book presents those results which can be proved using ZFC; and also includes an account of the necessary set-theoretic background, such as the notions of a regular cardinal and a stationary set. This is followed by a short intro- duction to set-theoretic forcing, which is aimed primarily at algebraists. The final three chapters explain why a number of natural problems concerning automorphism towers are independent of ZFC. In more detail, the book is organised as follows. In Chapter 1, we introduce the notion of the automorphism tower of a centreless group and illustrate this notion by computing the automorphism towers of a number of groups. In Chapter 2, we present a proof of Wielandt's theorem that the automorphism tower of a finite INTRODUCTION xi centreless group terminates after finitely many steps. In Chapter 3, we prove that the automorphism tower of an infinite centreless group G terminates after strictly + less than 2jGj steps. This chapter also contains an account of the basic theory of regular cardinals and stationary sets. Much of the remainder of this book is concerned with the problem of constructing centreless groups with extremely long automorphism towers. Unfortunately it is usually very difficult to compute the successive groups in the automorphism tower of a centreless group. In Chapter 4, we introduce the normaliser tower technique, which enables us to almost entirely bypass this problem. Instead, throughout most of this book, we only have to deal with the much easier problem of computing the successive normalisers of a subgroup H of a group G. As a first application of this technique, we prove that if κ is an infinite cardinal, then for each ordinal α < κ+, there exists a centreless group G of cardinality κ such that the automorphism tower of G terminates after exactly α steps. In Chapter 5, we present an account of Hamkins' work on the automorphism towers of arbitrary (not necessarily centreless) groups. Chapter 6 contains an introduction to set-theoretic forcing, which is intended to be intelligible to beginning graduate students. In Chapter 7, we show that it is impossible to find + a better bound in ZFC than 2jGj for the height of the automorphism tower of an infinite centreless group G. For example, it is consistent that there exists a centreless group G of cardinality @1 such that the automorphism tower of G has height strictly greater than 2@1 . On the other hand, in Chapter 9, we show that it is consistent that the height of the automorphism tower of every centreless group G of @1 cardinality @1 is strictly less than 2 . In Chapter 8, we consider the relationship between the heights of the automorphism towers of a single centreless group G computed in two different models M ⊂ N of ZFC. I am very grateful to David Nacin and Steve Warner for carefully reading the manuscript of this book and for supplying me with a substantial number of corrections. Thanks are due to the many friends and colleagues with whom I have discussed the material in this book, especially Greg Cherlin, Warren Dicks, Ulrich Felgner, Ed Formanek, Ken Hickin, Wilfrid Hodges, Otto Kegel, Felix Leinen, Peter Neumann, Dick Phillips and John Wilson. Particular thanks are due to my coauthors Joel xii INTRODUCTION Hamkins and Winfried Just. Of course, like so many logicians, my greatest debt of gratitude is owed to Saharon Shelah, who I would like to thank for all that he has taught me over the last twenty years.
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