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.

Restricted linear spaces

By the Fundamental Theorem (see Theorem 1.5.5) we know that b ≧ v for any finite linear space. Moreover, the linear spaces with b = v are precisely the (possibly degenerate) projective planes. It is natural to ask “What are the finite linear spaces satisfying b = v + 1, b = v + 2,…?” Indeed, these questions have been asked and answered in J. Totten (1975, 1976a, b).

In this chapter we want, however, to consider a much more general situation. Following Totten, we call a finite linear space restricted if it satisfies (b − v)2 ≦ v. The aim of this chapter is to classify all restricted linear spaces.

In order to formulate the results one particular notion is important. Let A be an affine plane, and denote by п1,…, пm some of its parallel classes. Denote by S' any linear space with point set {п1,…, пm}. Let S be the linear space

Then S is called an affine plane with the linear space S' at infinity.

For example, the Fano quasi-plane is the affine plane of order 2 with a near pencil at infinity (see Figure 6.1.1). (The affine plane of order 2 consists of the three “vertices” and the “midpoint” of the triangle.)

Introduction

One of the most natural strictly numerical questions to ask is what can be said if all the line degrees of a linear space S are known. Clearly, this problem will have a reasonable answer only if the set of allowable line degrees is quite small. If there is only one line degree, then S is a design, and in a sense, S is ‘known’. We therefore turn to the case of two line sizes. Work on two consecutive line sizes was the first to appear, and this was done by L. M. Batten, J. Totten and P. de Witte.

If we place an upper bound on v with respect to the line sizes, then we show below that we are able to say something quite precise about S in terms of its structure relative to a projective plane. The case of two line sizes includes the case of one line size. Thus we also give a precise description of designs with v points where v is bounded above. The results of Sections 3.2 and 3.3 are found in de Witte and Batten (1983) and in Batten and Totten (1980).

In Section 3.3 we present the work of Batten (1980) for three consecutive line sizes.

Section 3.4 deals with two non-consecutive line degrees; and in Section 3.5 we briefly describe some general theorems covering a broader class of line size problems.

Let bk and vr be the number of k-lines, respectively r-points, in S.

Definitions and examples

Definition: A linear space is a pair S = (p, ℒ) consisting of a set p of elements called points and a set ℒ of distinguished subsets of points, called lines satisfying the following axioms:

(L1) Any two distinct points of S belong to exactly one line of S.

(L2) Any line of S has at least two points of S.

(L3) There are three points of S not on a common line.

It is clear that (L3) could be replaced by an axiom (L3)': There are three lines of S not incident with a common point. In any case, (L3) and (L3)' are ‘non-triviality’ conditions. The readers should quickly describe those systems satisfying (LI) and (L2) but not (L3). These are called trivial linear spaces.

Points will usually be denoted by the lower case letters p, q, s,…, x, y, z, and lines by the upper case letters L, M, N,…, X, Y, Z.

The line through the distinct points p and q will be denoted by pq. If two distinct lines L and M intersect in some point, then their (unique) point of intersection will be denoted by L ∩ M.

We shall use ‘geometric’ language such as ‘a point is on a line’, ‘a line goes through a point’ and so forth, rather than confining ourselves to precise set-theoretic terminology.

Throughout this book we shall be restricting ourselves to finite linear spaces, that is, to linear spaces for which the point set is finite.

Abstract

Let Δ be the line graph of PG(n – 1, q), q > 2, Alt(n, q) be the graph of the n-dimensional alternating forms over GF(q), n ≥ 4. It is shown that every connected locally Δ graph, such that the number of common neighbours of any pair of vertices at distance two is the same as in Alt(n, q), is covered by Alt(n, q).

Introduction

There have been extensive studies in local characterization of graphs. Certain strongly regular graphs are characterized by their local structure. In this paper we shall investigate graphs which are locally a (q – l)-clique extension of the Grassmann graph over GF(q), q > 2. The Grassmann graph has as vertices all 2-spaces of an n-dimensional vector space V over GF(q). Two vertices are adjacent whenever they intersect nontrivially. The alternating forms graph Alt(n, q) is locally a (q – l)-clique extension of. In this paper, we restrict ourselves to the case μ = q2(q2 + 1), i.e., the number of common neighbours of two vertices at distance 2 is always q2(q2 + 1). Under the assumption μ = q2(q2 + 1), Alt(4, q) is the only graph which is locally (q – l)-clique extension of with n = 4. This result follows from the classification of affine polar spaces due to Cohen and Shult [2].

Abstract

If Γ is a thick and residually connected Dn-geometry, n ≥ 4, it is well known that Γ is denned over a unique ground division ring which is commutative. Here we give an elementary proof of the commutativity based on the construction of null polarities in the projective subspaces of Γ, for n = 4.

Introduction

Geometries over a diagram of type Dn, n ≥ 4 are completely classified and well known. They are often discussed together with polar spaces since they stem essentially from quadrics of maximal Witt index. However, the theory of polar spaces is rather intricate and it requires well over a hundred printed pages as we can see from [11] or [4]. A straightforward theory for Dn geometries is much simpler and shorter. Actually, it may come next to the theory of projective geometries by its simplicity. This was observed quite early ([10]).

We are developing a selfcontained theory of Dn-geometries from the definition to the classification. To the best of our knowledge such a treatment is not yet available. It may be useful in various directions, in particular the preparation of extensions to geometries over slightly more general diagrams.

The main result of the theory is as follows.

Abstract

We study arcs K in PG(n, q), n ≥ 3, q odd, having many points common with a given normal rational curve L. In particular, we show that, if 0.09q + 2.09 ≥ n ≥ 3, q large, then (q + l)/2 is the largest possible number of points of K on L, improving on the bound given in [11], [12], [14]. When |K ∩ L| = (q + l)/2, we show that the points of K ∩ L are invariant under a cyclic linear collineation of order (q ± l)/2. The corresponding questions for q even are discussed in [13].

Introduction

Let Σ = PG(n, q) denote the n-dimensional projective space over the field GF(q). A k-arc in Σ, with k ≥ n + 1, is a set K of k points such that no n + 1 points of K belong to a hyperplane of Σ. A point r of PG(n, q) extends a k-arc K, in PG(n, q), to a (k + l)-arc if and only if K ∪ {r} is a (k + l)-arc. A k-arc K of PG(n, q) is complete if and only if K is not contained in a (k + l)-arc of PG(n, q). Otherwise, K is called incomplete.

Abstract

We show that the Lyons group Ly has no distance-transitive representation, and that the only faithful multiplicity-free permutation representations of Ly are those on the cosets of G2(5) and of 3 M cL: 2.

One area which is still open in the classification of primitive distance-transitive graphs is the determination of the primitive distance-transitive representations of certain sporadic simple groups and their automorphism groups (see [1]). In this note we show that the Lyons group Ly ≅ Ant(Ly) has no distance-transitive representation. In the process, we find that the only faithful multiplicity-free permutation representations of Ly are those on the cosets of G2(5) and of 3 M cL:2.

Let G be a permutation group on a finite set Ω, and Γ a connected graph with vertex set Ω. (Throughout this note all graphs are undirected, with no loops and no multiple edges.) We say that G acts distance-transitively on Γ if for each i = 0,…, diam(Γ), G is transitive on the set of ordered pairs of vertices at distance i in Γ. The graph Γ is called distance-transitive if Aut(Γ) acts distance-transitively on Γ. A distance-transitive representation (DTR)

of a group X is a faithful permutation representation such that ρ(X) acts distance-transitively on some connected graph with vertex set Ω.

Let X be a finite group, and ρ : X → Sym(ω) a DTR.

Abstract

A nonic in PG(2, q), q = 2h, is defined to be either a non-degenerate conic plus its nucleus, or a degenerate conic, different from a repeated line, minus its nucleus. It is noted that every nonic is in the dual-line code of the plane, and so several questions arise. Can we have collections of nonics generating this code? Each oval (or (q + 2)-arc) is the sum (mod 2) of various numbers of nonics – what is the minimum number?

Introduction

There are four kinds of conics in a projective plane π := PG(2, q) over a finite field GF(q); see [5]. These are:

1. (1) irreducible conic, having precisely q + 1 points;

2. (2) distinct lines, having 2q + 1 points;

3. (3) lines in the quadratic extension PG(2, q2), intersecting in a point of π, and so having only one point;

4. (4) repeated line.

When q is even (and so q = 2h) the first three types of conies define a certain point called the nucleus, which lies on the conic if and only if the conic is reducible. In the first case it is the point of intersection of all the tangents; in the second it is the intersection of the two lines; in the third case the nucleus is the only point of π that is on the conic.

Abstract

We present a synthetical construction of the lifting process introduced in [1] and apply this process to obtain a new result on the structure of sets in the plane admitting the maximal number of nuclei.

Introduction

Let Bn be a set of qn−1 + qn−2 +…+ q + 1 points, not all on a hyperplane in the n-dimensional projective space PG(n, q) over the Galois field GF(q), n ≥ 2. A point not in Bn is called a nucleus of Bn if every line through it meets Bn (exactly once, of course). The set of all nuclei of Bn is denoted by N(Bn). The following two fundamental results are well known.

Result 1.1 (Segre-Korchmáros, [7]) If a, b, c are three non-collinear nuclei of Bn, then the points of Bn on the lines ab, bc, ca are collinear.

Result 1.2 (Blokhuis-Wilbrink, [3]) If Bn is an affine set (i.e. it is contained in the complement of a hyperplane), then |N(Bn)| ≤ q − 1.

The proofs of both the previous results have been given by the authors in the two dimensional case, but it is straightforward to see that they work in arbitrary dimensions. The original proof of Result 1.2 surprisingly does not use Result 1.1.

In the plane case, an elementary derivation of Result 1.2 from Result 1.1 has been obtained in [1] by using a process called “lifting”.

Abstract

Let Δ be the incidence graph of the unique biplane on 7 points, that is, the bipartite complement of the Heawood graph. We find that there are precisely three connected graphs that are locally Δ, on 36, 48 and 108 vertices, where the last graph is an antipodal 3-cover of the first one.

Introduction

Let notation be as in [1]. (In particular, ∼ denotes adjacency, Γi(γ) is the collection of vertices at distance i from γ in Γ, Γ(γ) := Γ1(γ), and γ⊥ := {γ} ∪ Γ(γ).) The Heawood graph H is the smallest cubic graph of girth 6; it is bipartite, the incidence graph of the Fano plane. The co-Heawood graph Δ is its bipartite complement, the nonincidence graph of the Fano plane, i.e., the incidence graph of the unique biplane on 7 points. (Thus, Δ = H3.) The graph Δ has 14 vertices, valency 4, is bipartite, is distance-regular of diameter 3 and has distance distribution diagram

Its automorphism group is G ≃ PGL(2, 7) of order 336 acting distance transitively.

The graph Δ occurs in the Suzuki chain S0 = 4K1, S1 = Δ, S2, S3, S4, S5 of graphs on 4, 14, 36, 100, 416, 1782 vertices, respectively. Each graph Si+1 of this chain is locally Si. In particular, the graph Σ := S2 is locally Δ, it is strongly regular with parameters (v, k, λ, μ) = (36,14,4,6).

Abstract

In this article, certain translation nets which are unions of subplane covered nets are shown to be equivalent to partially sharp subsets of PFZ(n, q). In particular, translation planes of order q2 and arbitrary kernel that admit two distinct Baer groups of order q–1 with identical component orbits are shown to be equivalent to partially sharp subsets of cardinality q of PTL(2, K) for some field K isomorphic to GF(q).

Introduction

Recently, a number of various connections have been established between translation planes and other geometric or combinatorial incidence structures. In particular, there are connections between flocks of hyperbolic quadrics in PG(3, q) and translation planes with spreads in PG(3, q) such that the spread is a union of reguli that share two lines. Similarly, partial flocks correspond to translation nets whose partial spreads are unions of reguli that share two lines.

It is well known that a Miquelian Minkowski plane can be defined using a hyperbolic quadric in PG(3, q) and when this connection is made, the points of the Minkowski plane are identified with the elements of PG(1, q) × PG(1, q), the circles (conies) correspond to the elements of PGL(2, q) and a flock corresponds to a sharply transitive subset of PGL(2, q).