Frank O. Wagner

Stable Groups

Groups & Guidance Table of Contents

Groups & Gist: Preface .......................................................... ix

Groups & Gratitude: Acknowledgements ............................ ix

0 Groups & Goals .................................................................... 1

0.0 Introduction

0.1 Getting to Grips with Groups

0.2 Mastering Minor Model Theory

0.3 Examples

0.4 Historical and Bibliographical Remarks

1 Groups & Generality ......................................................... 53

1.0 Chain Conditions

1.1 Connected Components and Definability

1.2 M_c-groups and Nilpotency

1.3 Substability and Local Conditions

1.4 Engel Conditions

1.5 Sylow Theory

1.6 Historical and Bibliographical Remarks

2 Groups & Genericity ....................................................... 107

2.0 The Superstable Case

2.1 Generic Types

2.2 Transitive Group Operations

2.3 Fields

2.4 Generic Properties

2.5 Historical and Bibliographical Remarks

3 Groups & Grandeur ........................................................ 142

3.0 Foreigness and Internality

3.1 Analysability and Groups

3.2 Components

3.3 Involutions

3.4 Groups without an Abelian Normal Subgroup

3.5 Localized Lascar Rank

3.6 Ranked Groups

3.7 Fields

3.8 Bad Groups

3.9 Historical and Bibliographical Remarks

4 Groups & Geometry ........................................................ 188

4.0 Pre-geometries

4.1 Local Modularity

4.2 Locally Modular Groups

4.3 The Group Configuration

4.4 Quasi-endomorphisms

4.5 CM-triviality

4.6 CM-trivial Groups

4.7 Dimensionality

4.8 Binding Groups

4.9 Historical and Bibliographical Remarks

5 Groups & Grades ............................................................. 250

5.0 R-groups

5.1 Abelian R-groups and R-fields

5.2 A Decomposition Theorem for R-groups

5.3 Linear Operations

5.4 Solubility and Nilpotence

5.5 Complements and Carter Subgroups

5.6 The Frattini Subgroup

5.7 Involutions and Conjugacy Classes

5.8 Historical and Bibliographical Remarks

Groups & Glory: References .............................................. 294

Groups & Gobbledegook: Index ........................................ 303

Errata et Addenda

Page 50

The group constructed by Baudisch in [11] is uncountably categorical, not totally categorical.

Corollary 3.6.12

A definably simple non-abelian type-definable group G of ordinal non-zero U_P-rank is simple.

It need not be definable, unless is is contained in a set X of ordinal U_P-rank.

Proof:

By stability, G is the intersection of definable supergroups. By compactness, one of them is contained in X and has ordinal U_P-rank; it follows that G is contained in a definable supergroup H of the same U_P-rank. P-connectivity of G and the Lascar inequalities for U_P imply that G is normal in HG is normal in H, so for any non-trivial g in G the conjugacy class g^H is definable and generates G definably. ¶

Simple Theories

Table of Contents

Preface .................................................................................. ix

Acknowledgements .............................................................. xi

1. Preliminaries .................................................................... 1

1.1 Introduction

1.2 Notation and model-theoretic prerequisites

1.3 Examples

1.4 Bibliographical remarks

2. Simplicity ....................................................................... 15

2.1 The monster model and imaginaries

2.2 Dividing and forking

2.3 Simplicity

2.4 Morley sequences

2.5 The Independence Theorem

2.6 Simplicity and Independence

2.7 Bounded equivalence relations

2.8 Types

2.9 Stability

2.10 Bibliographical remarks

3. Hyperimaginaries .......................................................... 51

3.1 Hyperimaginaries

3.2 Forking for hyperimaginaries

3.3 Canonical bases

3.4 Internality and analysability

3.5 P-closure and local modularity

3.6 Elimination of hyperimaginaries

3.7 The Lascar group

3.8 Bibliographical remarks

4. Groups ............................................................................ 95

4.1 Type-definable groups

4.2 Relatively definable groups

4.3 Hyperdefinable groups

4.4 Chain conditions and commensurativity

4.5 Stabilizers

4.6 Quotient groups and analysability

4.7 Generically given groups

4.8 Locally modular groups

4.9 Bibliographical remarks

5. Supersimple theories ................................................... 147

5.1 Ranks

5.2 Weight and domination

5.3 Elimination of hyperimaginaries

5.4 Supersimple groups

5.5 Type-definable supersimple groups

5.6 Supersimple division rings

5.7 Bibliographical remarks

6. Miscellaneous ............................................................... 187

6.1 Small theories

6.1.1 Elimination of hyperimaginaries

6.1.2 Locally modular theories

6.1.3 Theories with finite coding

6.1.4 Lachlan's conjecture

6.2 w-categorical theories

6.2.1 An amalgamation construction

6.2.2 w-categorical supersimple groups

6.2.3 w-categorical CM-trivial theories

6.3 Simple expansions of simple theories

6.3.1 Amalgamating simple theories

6.3.2 Simple theories with an automorphism

6.4 Low theories

6.5 Bibliographical remarks

Bibliography ...................................................................... 294

Index ................................................................................... 303

Errata et Addenda

Theorem 1.2.13

The correct bound for the Erdös-Rado Theorem is (2^2...2k)⁺ (a tower of n exponentiations), not (2^k)⁺.

Lemma 2.4.3

Since we don't know type-definability of independence yet, the compactness argument at the end of the proof is not quite straightforward: We obtain a chain of n-types p_n over A such that there are independent n-indiscernible sequences with that n-type of arbitrary length; by compactness there is an indiscernible sequence whose n-type is p_n for all n; it is independent by finite character.

Proposition 2.8.16

There is an example by Casanovas and Wagner satisfying conditions 2.-6. (which are equivalent by the proof given) but not condition 1.

Theorem 3.2.8

The proof of Local Character needs an additional argument:

Suppose A={a_i : i < k } is a set of countable hyperimaginaries. Choose inductively real tuples ã_i for i < k with ã_i independent over a_i of A U {ã_j : j < i } for all i < k. It is then easy to see that for all subsets A' of A, if Ã' = {ã : a in A'}, then Ã' is independent of A over A'. We can find a subset Ã₀ of à of small cardinality such that the partial type over A (represented as a partial type over Ã) does not fork over Ã₀. By Lemma 3.2.4 (or symmetry and transitivity) it does not fork over A₀ either, and Local Character holds. ¶

Lemma 3.4.19

The proof given is incomplete: the existence of a finite ã is not clear, as Proposition 3.4.9 uses the fact that the family of formulas is over A.

Sections 3.4 and 3.5

Note that internality and analysability are with respect to a class of partial types (which is hence closed under addition of parameters), whereas co-foreigness is with respect to a class of complete types (so we can only take non-forking extensions).

Proposition 3.7.10

Casanovas, Lascar, Pillay and Ziegler have a more convincing proof of part 2.

Lemma 3.6.12.3

Read tp(a) for tp(b).

Problem 3.7.21

M. Ziegler has constructed a theory which is not G-compact.

Definition 4.1.5 and Lemma 4.1.6

The rank D^* as given need not be translation invariant, as outside G we may lack associativity. Everything will work, however, if we choose a definable superset X of G such that multiplication is associative on X³, and consider only a_i with ø(x, a_i) contained in X.
Note that the h from Lemma 4.1.6 should be in G.

Lemma 4.1.9

In the proof of 2. ⇒ 3. the end of the displayed formula should read "-1" rather than "+1".

Proposition 4.2.7

Type-definability of the family H is not needed; it is enough to have a family of uniformly relatively definable subgroups of G satisfying the hypothesis of Lemma 4.2.6 (i.e. a subfamily of a type-definable family).

Theorem 4.5.13

In the definition of N (middle of page 126), one should not take the intersection over all a realizing p, but only over bounded sets I of realizations of p containing the sequence (a_i:i<ω), and then choose I such that the resulting group is minimal possible. It will have the required properties (A-invariant, hyperdefinable, between K and LK).

Section 4.7

Meanwhile, there has been significant progress concerning two of the most important technical tools one would like to have for simple theories: The group configuration theorem and the binding group theorem.

The group configuration theorem states that given a certain forking configuration, one can construct from it a canonical group action. Following the development of the theory of many-valued function germs by Itay Ben-Yaacov (Group configurations and germs in simple theories, JSL 67(4):1581-1600, 2002), the group was found by Ben-Yaacov, Tomasic and Wagner Constructing an almost hyperdefinable group; however it is living in the collection of almost hyperimaginaries (classes modulo invariant equivalence relations covered uniformly by boundedly many hyperdefinable sets). The group action was then recovered by Tomasic and Wagner Applications of the group configuration theorem in simple theories.

The binding group theorem says that if a type p is almost orthogonal to q^w but non-orthogonal to q, then the restriction to (the realizations of) p of the group of automorphisms (of the monster model) fixing q (and all parameters) is definable. In the simple context this group has been studied by Bradd Hart and Ziv Shami; however it may not properly reflect forking (adding a random bipartite graph between p and q will trivialize it). One may alternatively consider the group of elementary permutations of p over q (which is still trivialized by an added bipartite random graph, but not quite as bad), see Shami and Wagner. Using the group configuration theorem (and consequently almost hyperimaginaries), the group corresponding to the forking geometry was found by Ben-Yaacov and Wagner On almost orthogonality in simple theories.

A survey of these results has appeared in the Bulletin of Symbolic Logic.

Theorem 5.4.5

The condition SU(XH) < SU(H) + ω^α holds only for hyperdefinable X contained in a finite product of sets in the family X. The compactness argument that a hyperdefinable X contained in ⟨X⟩ is contained in a finite product of elements from X only works if the family X consists of definable sets. A counterexample even in the stable case is given by X=G a definable group, and X consisting of the connected component G° of infinite index together with a system of representatives for G/G°.
Moreover, the inequality is to be interpreted as SU(q) < SU(H) + ω^&alpha for all types q in XH.

Lemma 6.1.8

Third line of proof: tp(¯z ¯y / A ).

Example 6.2.27

We should note that the weight of R is 1, as in example 6.2.28.

Proposition 6.3.13 and 6.3.15

It should be T₁∪T₂ instead of T₁∩T₂.

Remark 6.4.2

False: There is an example of a simple theory with non-ordinal D(x=x, ø). The problem is that D(., ø) need not be continuous. The condition that D(x=x, ø) be ordinal for all formulas ø, called shortness, is studied by Casanovas and Wagner.

Lemma 6.4.4

In part 2., we have to require {ø(x,a_i) : i<w} to be inconsistent.

Frank O. Wagner Institut Camille Jordan et Institut universitaire de France