Introduction to Finite Geometries

North-Holland Texts in Advanced Mathematics: Introduction to Finite Geometries focuses on the advancements in finite geometries, including mapping and combinatorics.
The manuscript first offers information on the basic concepts on finite geometries and Galois geometries. Discussions focus on linear mapping of a given quadrangle onto another given quadrangle; point configurations of order 2 on a Galois plane of even order; canonical equation of curves of the second order on the Galois planes of even order; and set of collineations mapping a Galois plane onto itself. The text then ponders on geometrical configurations and nets, as well as pentagon theorem and the Desarguesian configuration, two pentagons inscribed into each other, and the concept of geometrical nets.
The publication takes a look at combinatorial applications of finite geometries and combinatorics and finite geometries. Topics include generalizations of the Petersen graph, combinatorial extremal problem, and theorem of closure of the hyperbolic space.
The book is a valuable source of data for readers interested in finite geometries.

Inhalt

Preface
Common Notation
Chapter 1 Basic Concepts concerning Finite Geometries
1.1 The Finite Plane
1.2 Isomorphic Planes, Incidence Tables
1.3 Construction of Finite Planes, Cyclic Planes
1.4 The R-Table of a Finite Projective Plane
1.5 Coordinate Systems on the Finite Plane
1.6 The Concepts of Galois Planes and Galois Fields
1.7 Closed Subplane of a Finite Projective Plane
1.8 The Notion of the Finite Affine Plane
1.9 Different Kinds of Finite Hyperbolic Planes
1.10 Galois Planes and the Theorem of Desargues
1.11 a Non-Desarguesian Plane
1.12 Couineations and Groups of Collineations of Finite Planes
1.13 Line Preserving Mappings of a Finite Affine Plane and of a Finite Regular Hyperbolic Plane
1.14 Finite Projective Planes and Complete Orthogonal Systems of Latin Squares
1.15 The Composition of the Linear Functions and the D(X Y) Plane
1.16 Problems and Exercises to Chapter 1
Chapter 2 Galois Geometries
2.1 The Notion of Galois Spaces
2.2 The Galois Space as a Configuration of Its Subspaces
2.3 The Generalization of Pappus' Theorem on the Galois Plane
2.4 Coordinates on a Galois Plane
2.5 Mappings Determined by Linear Transformations
2.6 Linear Mapping of a Given Quadrangle onto Another Given Quadrangle
2.7 The Concept of An Oval on a Finite Plane
2.8 Conies on a Galois Plane
2.9 Point Configurations of Order 2 on a Galois Plane of Even Order
2.10 The Canonical Equation of Curves of the Second Order on the Galois Planes of Even Order
2.11 Point Configurations of Order 2 on a Galois Plane of Odd Order
2.12 Correspondences between Two Pencils of Lines
2.13 a Theorem of Segre
2.14 Supplementary Notes concerning The Construction of Galois Planes
2.15 Collineations and Homographies on Galois Planes
2.16 The Characteristic of a Finite Projective Plane
2.17 The Set of Collineations Mapping a Galois Plane onto Itself
2.18 Desarguesian Finite Planes
2.19 Problems and Exercises to Chapter 2
Chapter 3 Geometrical Configurations and Nets
3.1 The Concept of Geometrical Configurations
3.2 Two Pentagons Inscribed into Each Other
3.3 The Pentagon Theorem and the Desarguesian Configuration
3.4 The Concept of Geometrical Nets
3.5 Groups and R Nets
3.6 Problems and Exercises to Chapter 3
Chapter 4 Some Combinatorial Applications of Finite Geometries
4.1 a Theorem of Closure of the Hyperbolic Space
4.2 Some Fundamental Facts concerning Graphs
4.3 Generalizations of the Petersen Graph
4.4 a Combinatorial Extremal Problem
4.5 The Graph of the Desargues Configuration
4.6 Problems and Exercises to Chapter 4
Chapter 5 Combinatorics and Finite Geometries
5.1 Basic Notions of Combinatorics
5.2 Two Fundamental Theorems of Inversive Geometry
5.3 Finite Inversive Geometry and the T-(v, k, ë) Block Design
5.4 General Theorems concerning the Möbius Plane
5.5 Incidence Structure and the /-Block Design
5.6 Problems and Exercises to Chapter 5
Chapter 6 Some Additional Themes in the Theory of Finite Geometries
6.1 The Fano Plane and the Theorem of Gleason
6.2 The Derivation of New Planes from the Galois Plane
6.3 a Generalization of the Concept of the Affine Plane
6.4 Problems and Exercises to Chapter 6

7. Appendix
   7.1 Notes concerning Algebraic Structures in General
   7.2 Notes concerning Finite Fields and Number Theory
   7.3 Notes concerning Planar Ternary Structures
   Bibliographical Notes
   Index