An Introduction to Finite Projective Planes (Dover Books on Mathematics) 0486789942, 9780486789941
Geared toward both beginning and advanced undergraduate and graduate students, this self-contained treatment offers an e
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English Pages [112]
Table of contents :
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
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
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