Introduction to Finite Geometries (eBook)

F. Karteszi (Autor)

eBook Download: PDF

2014 | 1. Auflage
280 Seiten
Elsevier Science (Verlag)
978-1-4832-7814-8 (ISBN)

Infos

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

Front Cover 1
Introduction to Finite Geometries 4
Copyright Page 5
Table of Contents 10
Dedication 6
PREFACE 8
COMMON NOTATION 13
CHAPTER 1. 
16 
1.1 The finite plane 16
1.2 Isomorphic planes, incidence tables 19
1.3 Construction of finite planes, cyclic planes 23
1.4 The G-table of a finite projective plane 27
1.5 Coordinate systems on the finite plane 34
1.6 The concepts of Galois planes and Galois fields 37
1.7 Closed subplane of a finite projective plane 42
1.8 The notion of the finite affine plane 45
1.9 Different kinds of finite hyperbolic planes 47
1.10 Galois planes and the theorem of Desargues 51
1.11 A non-Desarguesian plane 58
1.12 Collineations and groups of collineations of finite planes 62
1.13 Line preserving mappings of a finite affine plane 
69 
1.14 Finite projective planes and complete orthogonal systems of Latin squares 73
1.15 The composition of the linear functions and the D 
83 
1.16 Problems and 
93 
CHAPTER 2. 
99 
2.1 The notion of Galois spaces 99
2.2 The Galois space as a configuration of its subspaces 102
2.3 The generalization of Pappus' theorem on the Galois plane 109
2.4 Coordinates on a Galois plane 113
2.5 Mappings determined by linear transformations 121
2.6 Linear mapping of a given quadrangle onto another given quadrangle 122
2.7 The concept of an oval on a finite plane 125
2.8 Conies on a Galois plane 128
2.9 Point configurations of order 2 on a Galois plane of even order 134
2.10 The canonical equation of curves of the second order on the Galois planes of even order 137
2.11 Point configurations of order 2 on a Galois plane of odd order 140
2.12 Correspondences between two pencils of lines 143
2.13 A theorem of Segre 148
2.14 Supplementary notes concerning the construction of Galois planes 152
2.15 Collineations and homographies on Galois planes 157
2.16 The characteristic of a finite projective plane 158
2.17 The set of collineations mapping a Galois plane onto itself 161
2.18 Desarguesian finite planes 165
2.19 Problems and exercises to Chapter 2 177
CHAPTER 3. GEOMETRICAL CONFIGURATIONS AND NETS 179
3.1 The concept of geometrical configurations 179
3.2 Two pentagons inscribed into each other 181
3.3 The pentagon theorem and the Desarguesian configuration 185
3.4 The concept of geometrical nets 192
3.5 Groups and R nets 196
3.6 Problems and exercises to Chapter 3 198
CHAPTER 4. SOME COMBINATORIAL APPLICATIONS OF FINITE GEOMETRIES 200
4.1 A theorem of closure of the hyperbolic space 200
4.2 Some fondamental facts concerning graphs 202
4.3 Generalizations of the Petersen graph 206
4.4 A combinatorial extremal problem 210
4.5 The graph of the Desargues configuration 213
4.6 Problems and exercises to Chapter 4 214
CHAPTER 5. COMBINATORICS AND FINITE GEOMETRIES 216
5.1 Basic notions of combinatorics 216
5.2 Two fundamental theorems of inversive geometry 220
5.3 Finite inversive geometry and the t-(v, k, 
224 
5.4 General theorems concerning the Möbius plane 228
5.5 Incidence structure and the 
231 
5.6 Problems and exercises to Chapter 5 233
CHAPTER 6. SOME ADDITIONAL THEMES IN THE THEORY OF FINITE GEOMETRIES 234
6.1 The Fano plane and the theorem of Gleason 234
6.2 The derivation of new planes from the Galois plane 247
6.3 A generalization of the concept of the affine plane 256
6.4 Problems and exercises to Chapter 6 259
CHAPTER 
265 
7.1 Notes concerning algebraic structures in general 265
7.2 Notes concerning finite fields and number theory 272
7.3 Notes concerning planar ternary structures 276
BIBLIOGRAPHICAL NOTES 278
INDEX 280

Produkt-Details

• Erscheint lt. Verlag: 12.5.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• ISBN-10: 1-4832-7814-X / 148327814X
• ISBN-13: 978-1-4832-7814-8 / 9781483278148