An Introduction to Finite Projective Planes

Abraham Adrian Albert, Reuben Sandler

• Publisher: Dover Publications
• Year: 2014
• ISBN:
  9780486801353
• Topic: Matemáticas
• Subtopic: Geometría

Geared toward both beginning and advanced undergraduate and graduate students, this self-contained treatment offers an elementary approach to finite projective planes. Following a review of the basics of projective geometry, the text examines finite planes, field planes, and coordinates in an arbitrary plane. Additional topics include central collineations and the little Desargues' property, the fundamental theorem, and examples of finite non-Desarguesian planes.
Virtually no knowledge or sophistication on the part of the student is assumed, and every algebraic system that arises is defined and discussed as necessary. Many exercises appear throughout the book, offering significant tools for understanding the subject as well as developing the mathematical methods needed for its study. References and a helpful appendix on the Bruck-Ryser theorem conclude the text.

[1]

Elementary Results

1. Introduction

As in most areas of modern mathematics, the foundations of projective geometry can be formulated in the language of set theory. We shall begin the discussion of our subject from that point of view, since the fundamental concepts of set theory are familiar to almost everyone with recent training in elementary mathematics, and so will serve as a convenient vehicle for its introduction.

As the basic ingredients of our discussion, we shall require only the notion of a (finite or infinite) set Π of elements P₁, P₂, · · · , together with a (finite or infinite) collection of distinguished subsets

,

, · · · of Π, and the concept of inclusion. Thus, if P is an element of Π in a distinguished subset

of Π, we shall write

. If P is not an element of

, we write

.

We shall not be interested in every collection of subsets of every set, but shall concern ourselves only with those sets Π which, together with certain distinguished subsets

, satisfy the following three properties:

(a) If P and Q are any two distinct elements of Π, there is a unique distinguished subset

of Π such that

and

.

(b) If

and

are any two distinguished subsets of Π, there is a unique element P of Π such that

and

.

(c) There are four distinct elements P₁, P₂, P₃, P₄ of Π such that no three of them are elements of a common distinguished subset

of Π.

Before proceeding to our general study of those sets and collections of distinguished subsets that satisfy these three conditions we shall examine certain specific examples.

2. Examples of Systems

Our first example of a mathematical system that satisfies the properties just described begins with a set labeled Π₇ and which consists of seven elements P₁, P₂, P₃, P₄, P₅, P₆, P₇. We next select seven distinguished subsets

each consisting of three elements and defined as follows:

Our mathematical system consists of the set Π₇ and the distinguished subsets

and we can verify that our inclusion properties (a), (b), and (c) all hold in a straightforward manner. For example, we can verify that no pair of distinct elements P_i and Ρ_j belongs to two distinct subsets

and

, a result requiring (½)(7)(6) = 21 verifications, since there are 21 distinct pairs of points. Similarly the subset

contains the points P₁, P₂, P₅ and exactly one of these is in each of the subsets

. Thus we need 21 verifications to show that each of the 21 pairs of distinct distinguished subsets

and

has exactly one element common to both subsets. Finally, no three of the four distinct elements P₁, P₂, P₃,...