Introduction to Finite Geometries

PrefaceCommon NotationChapter 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 1Chapter 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 2Chapter 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 3Chapter 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 4Chapter 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 5Chapter 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 67. Appendix 7.1 Notes concerning Algebraic Structures in General 7.2 Notes concerning Finite Fields and Number Theory 7.3 Notes concerning Planar Ternary StructuresBibliographical NotesIndex