Finite Geometries

Frontmatter

1. Basic concepts

Abstract

This chapter is of a preliminary nature. In its first three sections, we introduce the general notions of incidence structure and incidence preserving map, and define incidence matrices, an important tool for the proof of many non-trivial results to be encountered later. We shall develop general theorems on finite incidence structures (many of them originally proved under unnecessarily restrictive hypotheses) and thus set up the framework for the more specialized investigations in later chapters.

Peter Dembowski

2. Designs

Abstract

In this chapter we shall be concerned with the general theory of designs (these were defined in Section 1.1, and the definition will be repeated in 2.1). The discussion will by no means be complete: we shall concentrate mostly on results and problems significant for projective, affine, and inversive planes. These three types of designs are the subject matter of Chapters 3–6 below; only a few facts about them will be mentioned here.

Peter Dembowski

3. Projective and affine planes

Abstract

This chapter and the following two are in a relationship similar to the three parts of Chapter 2: the combinatorial part will be covered here automorphisms (collineations) follow in Chapter 4, and constructions in Chapter 5.

Peter Dembowski

4. Collineations of finite planes

Abstract

In this chapter we give a fairly complete account of the known results on automorphisms of finite projective and affine planes. As before, we shall use the term “collineation” rather than “automorphism”.

Peter Dembowski

5. Construction of finite planes

Abstract

The only finite projective and affine planes that we have actually defined so far (cf. Section 1.4) are the desarguesian planes P(q) and A(q). We have mentioned repeatedly that there exist non-desarguesian finite planes as well, and in this chapter we shall present all known such planes. The known construction techniques for finite planes all use a finite vector space in a more or less obvious, but always essential way. This is the reason that these constructions always lead to planes of prime-power order. It is one of the major unsolved problems of the theory whether or not there also exist planes of non-prime-power order.

Peter Dembowski

6. Inversive planes

Abstract

It is the object of this chapter to collect the known facts on finite inversive planes, which were defined in Section 2.4 as designs of type (1, 3) satisfying (2.4.19). We shall give a more geometric definition here, which is meaningful also in the infinite case. In fact, Section 6.1 (which may be compared with Section 3.1 on projective and affine planes) is concerned with inversive planes in general; finiteness will usually not be essential there.

Peter Dembowski

7. Appendices

Abstract

In this last chapter we will be concerned with four topics which, for one reason or another, do not fit naturally into the general setup of the earlier parts of the book. There are nevertheless enough connections to justify the inclusion of these subjects here.

Peter Dembowski

Backmatter