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.
Author(s): Abraham Adrian Albert, Reuben Sandler
Publisher: Dover Publications; Reprint edition
Preface
Contents
Chapter 1. Elementary Results
1.1 Introduction
1.2 Examples of Systems
1.3 Projective Planes
1.4 Subplanes
1.5 Incidence Structures
1.6 Isomorphism of Planes
1.7 Duality
1.8 The Principle of Duality
1.9 Desargues' Configuration
Chapter 2. Finite Planes
2.1 Introduction
2.2 Counting Lemmas
2.3 The Order of a Finite Plane
2.4 Loops and Groups
2.5 Collineations
2.6 The Incidence Matrix
2.7 Combinatorial Results
Chapter 3. Field Planes
3.1 Fields
3.2 Prime Fields
3.3 Field Planes
3.4 Matrices and Collineations of PG(2, p^n)
3.5 Analytic Geometry-Coordinates
Chapter 4. Coordinates in an Arbitrary Plane
4.1 Naming the Points and Lines
4.2 The Planar Ternary Ring
4.3 Further Properties of (R, F)
4.4 Collineations and Ternary Rings
Chapter 5. Central Collineations and the Little Desargues' Property
5.1 Central Collineations
5.2 Little Desargues' Property
5.3 Coordinatization Theorems
Chapter 6. The Fundamental Theorem
6.1 Coordinates in a Field Plane
6.2 Wedderburn's Theorem
6.3 The Fundamental Theorem
6.4 Pappus' Property
Chapter 7. Some Non-Desarguesian Planes
7.1 Subfields and Automorphisms of Finite Fields
7.2 The Algebras
7.3 A Concrete Example
Appendix-The Bruck-Ryser Theorem
References
Index
