MONOTONICITY

In this chapter, I consider the problem of determining which sequences of natural numbers can occur as (fn) or (Fn) for some oligomorphic permutation group. The problem in this generality is quite out of reach. All I can do is to illustrate by examples some of the possibilities that can occur, and to describe some of the restrictions on realisable sequences which have been established.

We saw in the last chapter that any permutation group is a dense subgroup of the automorphism group of a homogeneous relational structure, so it suffices to consider these groups. Moreover, since the group is oligomorphic, the structure is ℵ0-categorical. The results of §2.5 imply that the sequences (fn) and (Fn) realised by oligomorphic groups are precisely those which enumerate unlabelled and labelled structures respectively in a class of finite structures satisfying Fraïssé's hypotheses (notably, the amalgamation property). From this point of view, two subcases commend themselves to us:

1. (a) classes having the strong amalgamation property;

2. (b) ages of homogeneous structures over finite relational languages.

3. Other subcases are obtained by imposing model-theoretic conditions in the neighbourhood of stability on the homogeneous structure.

The most important basic restriction on realisable sequences is that they must be monotonic.

JORDAN GROUPS

Let G be a permutation group on Ω. A Jordan set for G is a subset of Ω with the property that the pointwise stabiliser of its complement acts transitively on it. (Sets consisting of just one point satisfy this condition trivially but are usually excluded for technical reasons.) If G is n-transitive, then any set containing all but n – 1 points of Ω is a Jordan set; such Jordan sets are called improper. (This needs some care in the case when n is infinite.) Then G is called a Jordan group if it has a proper Jordan set (other than the empty set).

With the exception of some recent examples constructed by Hrushovski (to appear), the known infinite Jordan groups are of three types:

(J1) Geometric examples: These are the projective group PGL(n, k),the affine group AGL(n, k), and their close relatives. The pointwise stabiliser of any subspace of a projective or affine space acts transitively on its complement. So the complements of subspaces are the Jordan sets, and the geometry can be recovered from them. In this class, it is customary now to include also the automorphism groups of algebraically closed fields (which preserve the geometry of algebraically closed subflelds). In each of these cases, the subspaces of the geometry are precisely the algebraically closed sets (in the sense of §2.7). This fact is crucial, both in their study, and in applications.

BEGINNINGS

This chapter is about subgroups of automorphism groups of various structures.

There are several aspects to note. For a start, as we saw in Chapter 2, if a group G is the automorphism group of a structure, then it is the automorphism group of a homogeneous structure; and this is equivalent to G being a closed subgroup of the symmetric group. Describing all subgroups of the symmetric group is too wide a task, so I'll restrict both the structures, and the kinds of subgroups considered. As to the first, I shall consider only

1. (a) ℵ0-categorical structures (those whose automorphism groups are oligomorphic); and

2. (b) homogeneous structures whose age has the strong amalgamation property (those for which the stabiliser of a tuple in the automorphism group fixes no additional points).

Similar results hold in other cases; a notable example of this is provided by recursively saturated structures. (The results for these structures, due to Richard Kaye, were obtained following the Durham symposium.)

The results will in the main be constructions of subgroups with various properties, but at the end of the chapter I will describe some restrictive results characterising certain kinds of subgroups (normal subgroups, subgroups of small index, etc.).

There are differing levels of detail about a subgroup. We could be concerned with its structure as abstract group, as permutation group on some subset of the domain, or as permutation group on the entire domain.

HISTORY AND NOTATION

In the summer of 1988, a London Mathematical Society symposium was held in Durham on “Model Theory and Groups”, organised by Wilfrid Hodges, Otto Kegel and Ileter Neumann. This volume of lecture notes is based on the series of lectures I gave at the symposium, but is something more: since no Proceedings of the symposium was published, I have taken the opportunity to incorporate parts of the talks given by other participants, especially David Evans, Udi Hrushovski, Dugald Macpherson, Ileter Neumann, Simon Thomas and Boris Zil'ber. (A talk by Richard Kaye revealed new horizons to me which I have not fully assimilated; but Richard's own book should appear soon.) In addition, I have made use of parts of the proceedings of the Oxford–QMC seminar on the same subject which ran weekly in 1987–8 and continues once a term (now as the Oxford–QMW seminar!); contributions by Samson Adeleke, Jacinta Covington, Angus Macintyre and John Truss have been especially valuable to me.

Why model theory and groups? In particular, why the special class of permutation groups considered here?

In the middle 1970s, when my interests were entirely finite, John McDermott asked a question about the relationship between transitivity on ordered and unordered n-tuples for infinite permutation groups. The analogous question, and more besides, had been settled for finite permutation groups by Livingstone and Wagner (1965), with techniques which were largely combinatorial and representation-theoretic, and so not likely to be useful here.

The Four Color Problem

When we have a polygonal map before us, we may think of the faces as being countries or states on a map, with the ocean surrounding them in the form of the infinite face. In a good atlas the countries, together with the ocean, are colored in different colors to distinguish them from each other. This means that the coloring must be done so that countries with a common boundary have different colors. If one has a large number of colors at one's disposal, this represents no particular problem. Much more difficult is the question of finding the smallest number of colors sufficient for coloring the countries of a given map.

A famous problem is to prove that every map can be colored properly by means of four colors. The earliest trace of it we find in a letter of 23 October 1852 from the London mathematician Augustus De Morgan to Sir William Rowan Hamilton in Dublin: “A student of mine asked me today to give him a reason for a fact which I did not know was a fact and do not yet. He says that if a figure be anyhow divided and the compartments differently coloured, so that figures with any portion of common boundary line are differently coloured—four colours may be wanted but no more.”

The term “graph” in this book denotes something quite different from the graphs you may be familiar with from analytic geometry or function theory. The kind of graph you probably have dealt with consisted of the set of all points in the plane whose coordinates (x, y), in some coordinate system, satisfy an equation in x and y. The graphs we are about to study in this book are simple geometrical figures consisting of points and lines connecting some of these points; they are sometimes called “linear graphs”. It is unfortunate that two different concepts bear the same name, but this terminology is now so well established that it would be difficult to change. Similar ambiguities in the names of things appear in other mathematical fields, and unless there is danger of serious confusion, mathematicians are reluctant to alter the terminology.

The first paper on graph theory was written by the famous Swiss mathematician Euler, and appeared in 1736. From a mathematical point of view, the theory of graphs seemed rather insignificant in the beginning, since it dealt largely with entertaining puzzles. But recent developments in mathematics, and particularly in its applications, have given a strong impetus to graph theory. Already in the nineteenth century, graphs were used in such fields as electrical circuitry and molecular diagrams.