Compact operators [61, X.5; 46, VI.5]. Let H be a Hilbert space (with a countable basis), ( , ) the scalar product on H, and ℒ(H) the algebra of bounded linear operators on H. If A ∈ ℒ(H) then A* denotes its adjoint – that is, the unique bounded linear operator such that (Ax, y) = (x, A*y) (x, y ∈ H). The operator A*A is self-adjoint and positive (i.e., (A*Ax, x) ≥ 0 for all x ∈ H) ; it has a unique positive square root, called the absolute value |A| of A.

The bounded operator A is compact if it transforms any bounded set into a relatively compact one. The eigenspaces of A corresponding to nonzero eigenvalues are finite dimensional. If A is compact, positive, and self-adjoint, then H has an orthonormal basis {ei} consisting of eigenvectors of A: Aei = λi,ei, with λi → 0. The operator A is compact if and only if |A| is. The compact operators obviously form an ideal in ℒ(H).

More generally, a continuous linear map A: H → H′ of H into a Hilbert space H′ is said to be compact if it transforms any bounded set into a relatively compact one or, equivalently, any bounded sequence into one containing a convergent subsequence.

In section 0, we shall introduce the various chain conditions which are implied by stability: the ascending and the descending chain conditions for intersections of uniformly definable families of subgroups (these intersections will then also be uniformly definable), and the ∣T∣+-chain condition for arbitrary Λ-definable subgroups. As we shall want these conditions to hold in saturated models as well, they are necessarily uniform. We shall also mention chain conditions which hold under stronger stability assumptions: the descending chain condition for definable subgroups in ω-stable groups, the descending chain condition for connected Λ-definable subgroups in the superstable case, and the ascending chain condition for connected definable subgroups in theories of finite rank.

However, chain conditions are important in their own right, not only in connection with stabilitYi once they are established, no further use of stability will be made in that chapter (with the exception of Theorem 1.1.13, which has been included in this section because of its proximity to the nilpotency results there). In fact, a more algebraic setting might consist of a group together with a family of sections comprising all centralizers and closed under taking normalizers, quotients, and – in the case of an abelian group – the subgroups of n-divisible elements and elements of order n for all n < ω; we would then require the uniform chain condition on intersections of uniform subfamilies.

In this chapter, we shall analyse the dependence relation associated with forking. After some remarks in section 0 about geometries and pre-geometries in general and the geometry of forking in particular, we add additional conditions on the family P of types whose forking geometry we want to study. In sections 1 and 2 this is local modularity; put crudely, a locally modular theory should essentially behave either like a set with no structure (the trivial case), or like a module over some ring. In section 1 the general theory of local modularity is developed, and it is then applied to groups in section 2: if the generic type of a group is locally modular, then the group is abelianby-finite; if the generic type is analysable in a locally modular family P of types, then the P-connected component of the group is nilpotent. In the case where the generic type of a group is locally modular and regular, we shall show that the module structure is actually present: generically the group behaves like a vector space over a certain division ring, namely the ring of quasi-endomorphisms. A particular case of local modularity is one-basedness; we shall show that a group is one-based iff it is an abelian structure.

Section 3 introduces an important tool in obtaining a definable group from structural considerations, the group configuration. We shall show in particular that a locally modular non-trivial family P of types gives rise to a type-definable P-semi-regular group acting faithfully and transitively on some set X, and we classify the possibilities for that action in the case Up(X) = 1. In particular, every locally modular regular type is either trivial (and so its forking geometry is uninteresting) or equivalent to the generic type of a group (and we may apply the results of the preceding section).

In this chapter we shall develop the second main tool for analysing stable groups, the method of generic types. In section 0 we shall develop the relevant notions for the case of a superstable group before introducing them in full generality in section 1, which culminates in a general existence and definability theorem for stable structures: any semi-group with a certain cancellation condition is contained in a definable group. The next, short, section generalizes the results obtained for groups to transitive group actions. We shall see the first simple applications in section 3, where we look at fields and prove equality of the additive and the multiplicative generic types. Furthermore we shall transfer the definability results to fields.

In section 4 we shall look at generic properties. This is inspired by algebraic group theory: if some equation is satisfied generically in an algebraic group, then it is satisfied by the whole group, as it defines a Zariski closed set of maximal dimension. For stable groups similar questions are mostly open; the only settled cases deal with nilpotency, solubility, and exponent 2 or 3. If the group is soluble-by-finite, then generically finite exponent implies finite exponent, and generically prime exponent implies that the group is nilpotent-by-finite.

The Superstable Case

Before we start with the general theory, we shall illuminate the relevant concepts in the superstable case. Readers familiar with this part of the theory may safely continue with section 1. Throughout this section G will denote a superstable group.

Introduction

The study of stable groups began with the classification of uncountably categorical abelian groups [69] and fields [70] by Angus Macintyre. In 1965 Michael Morley [75] had proved his Categoricity Theorem and defined wstability on the way, and Saharon Shelah [102, 103] embarked on his study of the models of complete first-order theories using his new notion of “stability”; these logical constraints were now being applied to algebraic structures. Differentially closed fields were studied by Shelah in [104], minimal groups by Joachim Reineke in [99], and finally simple groups by Gregory Cherlin in [34], where he formulated his famous conjecture: a simple group of finite Morley rank is an algebraic group over an algebraically closed field. Like Boris Zil'ber's conjecture from [134], of which it may be considered an analogue for groups, it is more programme than claim: one should try to do algebraic geometry, and in particular to define the Zariski topology, by model-theoretic means. Although the original conjecture was refuted by a counter-example of Ehud Hrushovski [57], the programme itself was completed successfully in [59] and has already led to the first applications outside of logic in Hrushovski's proof of the relative version of the Mordell-Lang Conjecture [56].

Meanwhile people had started to look at more general stability classes: Cherlin and Shelah were studying superstable division rings [33], fields and groups [37], Chantal Berline and Daniel Lascar [16, 17, 18] generalized many results from stable to superstable groups, and John Baldwin and Jan Saxl analysed stable groups in [4].

In this chapter we shall analyse stable groups with the additional property ℜ, a property common to small stable and superstable groups. We shall define it in section 0 and show that a small stable group is ℜ; the proof also implies that the index of the image of an endomorphism of a small group is at most the cardinality of the kernel (as long as both are finite). We then derive some immediate consequences like the independence of the generic type of the group law. In section 1 we shall prove that an infinite ℜ-group must always contain an infinite abelian subgroup, and look at ℜ-fields: an ℜ-ring without zero-divisors is a commutative, algebraically closed field, and any definable automorphism is ∅-definable and has finite fixed field.

In section 2 we shall prove an analysability result reminiscent of Remark 3.6.2, replacing regular types by abelian subgroups. In particular there exist abelian subgroups which are big, i.e. not co-foreign to the generic type of the whole group.

The next section deals with linear operations and the existence of definable fields. The problem here is one of comparing groups: if G operates on A as a group of automorphisms, in order to obtain a field, we have to take care that the sums Σi(giA) remain bounded, for elements gi from G. In a certain way this means that A and G have comparable size; technically it is achieved by requiring A to be G-minimal and gen (A) not to be foreign to G.

In this chapter model theory will play a more prominent role. In the noughth section we shall lay the model-theoretic foundations for the subsequent chapters and introduce the notions of “foreign” and “internal” due to Hrushovski. This allows us to generalize the concept of “small” (e.g. finite) and “large” (e.g. generic) sets: a definable subset is “small” if the generic type is foreign to it, and “large” if the generic type is internal or analysable in it. It will turn out in section 1 that these notions are particularly well-behaved in the context of groups: if the generic type of a group is not foreign to some set X, then there is a definable X-internal quotient of the group. This means that quite often it is sufficient to consider only definable subsets which define groups. Furthermore there is some form of compactness: if a group is internal in a class of definable sets, then it is internal in a subset of them, of bounded cardinality.

In section 2, we shall consider various components of a stable group. In particular we shall define the Φ-component, which enjoys very strong connectivity properties. It is the intersection of all relatively definable subgroups not only of finite, but of “small” index (defined in terms of analysability of the whole group using the quotient), and it generalizes both the characteristic normal subgroup of monomial U-rank given by Berline and Lascar, and Hrushovski's p-connected component (where p is a regular type in which the group is internal).

Abstract

We survey some recent classification theorems for elation generalized quadrangles of order (q2, q), q even, with particular emphasis on those involving subquadrangles of order q.

Abstract

If x is a regular point of a generalized quadrangle S = (P, B, I) of order (s, t), s ≠ 1, then x defines a dual net with t + 1 points on any line and s lines through every point. If s ≠ t, s > 1, t > 1, then S is isomorphic to a T3(O) of Tits if and only if S has a coregular point x such that for each line L incident with x the corresponding dual net satisfies the Axiom of Veblen. As a corollary we obtain some elegant characterizations of the classical generalized quadrangles Q(5, s). Further we consider the translation generalized quadrangles S(p) of order (s, s2), s ≠ 1, with base point p for which the dual net defined by L, with p I L, satisfies the Axiom of Veblen. Next there is a section on Property (G) and the Axiom of Veblen, and a section on flock generalized quadrangles and the Axiom of Veblen. This last section contains a characterization of the TGQ of Kantor in terms of the Axiom of Veblen. Finally, we prove that the dual net defined by a regular point of S, where the order of S is (s, t) with s ≠ t and s ≠ 1 ≠ t, satisfies the Axiom of Veblen if and only if S admits a certain set of proper subquadrangles.

Abstract

In the last few years there has been rapid progress in the theory of difference sets. This is a survey of these fascinating new developments.

Abstract

The ternary codes associated with the five known biplanes of order 9 were examined using the computer language Magma. The computations showed that each biplane is the only one to be found among the weight-11 vectors of its ternary code, and that none of the biplanes can be extended to a 3-(57,12,2) design. The residual designs of the biplanes, and designs associated with {12; 3}-arcs were also examined.

Abstract

It is shown that the function field of the algebraic envelope of the cyclic (q — √q +1) –arc in PG(2, q), q square, is the Hermitian function field.

Abstract

We consider the following problem: given a partial geometry with v points and k points on a line, can one add to the line set a set of k-subsets of points such that the extended family of k-subsets is a 2-(v, k;, 1) design (or a Steiner system S(2, k, v)). We give some necessary conditions for such embeddings and several examples. One of these is an embedding of the partial geometry PQ+(7,2) into a 2-(120,8,1) design.