Introduction

In this chapter we consider semiaffine linear spaces of a much more general nature than in the previous chapter. Also, our methods are quite different. We take a somewhat graph-theoretic approach, using terminology and some basic notions from that area. For us, the vertices of the graph are the lines of the linear space.

Our principle result (Theorem 5.3.3) can be interpreted as follows. If the order n of the linear space S is large enough with respect to the difference between the maximum point degree and the minimum line degree in S, then S embeds in a projective plane of order n. This chapter is based on Beutelspacher and Metsch (1986, 1987).

Our first result is a very general one. The proof is broken into several stages by using propositions. For lines L and H, we denote by m(L,H) the number of lines missing both L and H.

Theorem 5.1.1. Let S = (p,ℒ) be a linear space. Suppose there are a line H and integers a, c, d, e, n and x with the following properties.

1. (1) The degree of H is n + 1 − d > 0.

2. (2) The number of lines missing H is nd + x > 0.

3. (3) For every line L missing H we have n − 1 + a ≤ m(L, H) ≤ n − 1 + c.

4. (4) For any two intersecting lines L1 L2 missing H, we have m(L1,L2) ≤ e + 1.

5. […]

The definition

Given an arbitrary linear space S = (p, ℒ), there is a natural way of introducing further structure on S by distinguishing subsets of p with the property that any element of ℒ containing at least two points of such a subset must be wholly contained within the subset. We call any subset of p with the above property, along with the induced lines, a subspace of 5. Clearly φ and S themselves are subspaces. Moreover, the intersection of any set of subspaces is again a subspace (the empty intersection being S), and so a closure space is induced on S, the closure of any set X of points of p being the intersection of all subspaces containing X.

There are various ways in which a ‘dimension’ can now be assigned to subspaces. In any case, we normally wish to assign dimensions 0 and 1 to the points and lines respectively, and require that V ⊂ W implies that the dimension of V be less than or equal to the dimension of W. We shall make the following definition.

The following pages contain descriptions of the linear spaces on at most nine points. For each linear space, we give a picture, the number of lines, the collineation group acting on the space (this appears in a box) and the number of point orbits under this group.

We use the following group theoretic notations for the collineation groups:

Cn is the cyclic group of order n,

Sn is the symmetric group on n letters,

D2n is the dihedral group of order 2n,

пiH is the direct product of / groups H,

ΣiH is the direct sum of / groups H,

Gn is a group of order n.

The general notation G is used when the group is not describable as a direct sum or product of the cyclic, symmetric or dihedral groups.

We wish to thank Jean Doyen, who, with much time and effort, compiled this comprehensive table for us and offered to let us use it as an appendix to this monograph. Any errors in it are due to the present authors.

I-Semiaffine linear spaces

If (p,L) is a non-incident point-line pair, we denote by π (p,L) the number of lines through p which miss L. Using this notation, a linear space containing a quadrangle is a projective plane if and only if π (p,L) = 0 for every non-incident point-line pair. Likewise, a linear space containing a triangle is an affine plane if and only if π (p,L) = 1 for every non-incident point-line pair.

G. Pickert (1955) was the first to ask a natural generalization of these facts: What non-trivial linear spaces are characterized by

π(p,L) ≤ 1

for every non-incident point-line pair?

In 1962, P. Dembowski gave a complete description of all such finite linear spaces, which he proceeded to call semiaffine planes. (Hence, the term ‘semiaffinity condition’ used in Section 1.4.) As P. Dembowski mentions in his 1962 paper, N. Kuiper had at about the same time, and independently, proved the same result. It is therefore now usually referred to as the Kuiper–Dembowski theorem. The aim of the first section of this chapter is to prove this result.

In subsequent sections we shall make yet a further generalization: Let I be a set of non-negative integers. S is said to be I-semiaffine if π (p,L) ∈ I for every non-incident point-line pair (p,L) of S. S is said to be I-affine if S is I-semiaffine but not J-semiaffine for any proper subset J of I.

Aim of the chapter

The aim of this chapter is to show explicitly how certain linear spaces can be embedded in a projective plane. Among such structures are the complements of two lines, of a triangle, of a hyperoval and of a Baer subplane. Here, the notion of a pseudo-complement is crucial. Suppose that we remove a set X of a projective plane P of order n. Then we obtain a linear space P-X having certain parameters (i.e., the number of points, the number of lines, the point- and line-degrees). We call any linear space which has the same parameters as P-X a pseudo-complement of X in P.

We have already encountered the notion of a pseudo-complement, namely the pseudo-complement of one line. This is a linear space with n2 points, n2 + n lines in which any point has degree n + 1 and any line has degree n. We know that this is an affine plane, which is a structure embeddable into a projective plane of order n. Another example may help to clarify the above definition. A pseudo-complement of two lines in a projective plane of order n is a linear space having n2 − n points, n2 + n − 1 lines in which any point has degree n + 1 and any line has degree n − 1 or n. (More precisely, the n − lines form a parallel class.)

Reasons for this chapter and some background

Up to this point in the book, the results we have presented are aimed at finding relationships between linear spaces and projective spaces. In this chapter we consider the broader question of how linear spaces relate to each other. This is too general a problem to tackle with our earlier methods. Our approach will be to introduce groups. This is a fairly new approach, with relevant articles first appearing perhaps twenty years ago. A large part of the research has been aided by recent important developments in group theory – in particular, the classification of finite simple groups. We shall list in this section those group-theoretic results needed for the chapter. We give them without proof, but of course refer to articles or books in which proofs may be found.

We emphasize the fact that there are many results about groups operating on linear spaces, which we shall not include here, because the results are about the group structure rather than the linear space structure.

The first major result, to which we shall refer later, is the classification of all finite 2-transitive groups. This classification is a consequence of the classification of all finite simple groups. For the latter, we refer the reader to D. Gorenstein (1982). The result we give here can be found in W. M. Kantor (1985).

Theorem 8.1.1. Let G be a 2-transitive group of permutations of a finite set X on v elements.

The aim of this monograph is to give a comprehensive and up-to-date presentation of the theory of finite linear spaces. For the most part, we take a combinatorial approach to the subject, but in the final chapter group theory is introduced.

The text is designed as a research resource for those working in the area of finite linear spaces, while the structure of the book also encourages its use as a graduate level text. At the end of each chapter, there is a section of exercises designed to test and extend a student's knowledge of the material in that chapter. There is also a research problem section containing current open problems which can be tackled with the aim of producing a thesis or a journal publication.

In the first chapter, constructions of affine and projective spaces are reviewed, and the fundamental results on finite linear spaces are given. Chapters 2 through 6 cover the work done on the major problem areas in linear spaces taking the ‘planar’ view: classification of linear spaces with given parameters, embeddability of linear spaces in “suitably small” projective planes. In Chapter 7 we consider problems of embedding higher dimensional linear spaces in projective spaces. Finally, in Chapter 8, assumptions are introduced on the collineation groups of linear spaces, and the recent results on characterization are presented.

There are several people we wish to thank for their assistance, encouragement and patience while this book was being written.