Introduction

Summary

One of the features of geometry, and of finite geometry in particular, is the difficulty of giving a concise definition of the subject. As well as the wide variety of structures that are studied and techniques that are used, an important factor contributing to this intractibility is the way in which different parts of the subject link up with and influence one another. This is part of the excitement of the subject for its practitioners, but may be off-putting for outsiders who see a confused tangle rather than an elegant network. The purpose of this introduction is to attempt to trace some of the main threads of finite geometry, and to locate the papers of this collection in the warp and weft of its fabric.

The structure of the subject militates against a linear tour of its highlights; but, of course, there is no other way to write an introduction. To simplify the task, we regard finite projective geometries “Galois spaces” as the central concept.

Let n be a positive integer, and q a prime power; let GF(q) denote the Galois field with q elements. The elements of the n-dimensional projective geometry PG(n,q) or S_n,q are the subspaces of an (n+1)-dimensional vector space V over GF(q); each has a geometric dimension which is one less than its vector space dimension. Thus the basic objects, the points, are the 1-dimensional subspaces of V. It is common to identify an arbitrary subspace with the set of points it contains.