Finite Geometries

Gyorgy Kiss, Tamas Szonyi

• Publisher: Chapman and Hall/CRC
• Year: 2019
• Print ISBN: 9781032475387
• eBook ISBN: 9781351646383
• Edition: 1
• Topic: Mathematics
• Subtopic: Counting & Numeration
• Index: Mathematics

Finite Geometries stands out from recent textbooks about the subject of finite geometries by having a broader scope. The authors thoroughly explain how the subject of finite geometries is a central part of discrete mathematics. The text is suitable for undergraduate and graduate courses. Additionally, it can be used as reference material on recent works.

The authors examine how finite geometries' applicable nature led to solutions of open problems in different fields, such as design theory, cryptography and extremal combinatorics. Other areas covered include proof techniques using polynomials in case of Desarguesian planes, and applications in extremal combinatorics, plus, recent material and developments.

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• Includes exercise sets for possible use in a graduate course

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• Discusses applications to graph theory and extremal combinatorics

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• Covers coding theory and cryptography

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• Translated and revised text from the Hungarian published version

1

Definition of projective planes, examples

In this book some familiarity with classical geometry will be assumed. The classical results will not be used explicitly, but will just provide some background motivation for some of the results. Probably everyone has learnt about Euclidean planes. The classical projective plane comes from the classical Euclidean plane by introducing ideal (or infinite) elements. Associated to a parallel class of lines we have an ideal (or infinite) point, and the ideal line (or line at infinity) consists of all the infinite points. The advantage of introducing the classical projective plane is that there is no difference between ordinary and ideal points; two lines always intersect. In classical geometry typical theorems state that under some conditions certain lines pass through a point (for example, if we take a triangle, then the angle bisectors pass through a point) or certain points are on a line. In some cases, the classical theorems use metric properties of the plane (distances and angles), in other cases the order of the points on a line, but there are interesting results that just use incidences of points and lines. A notable example for this is the celebrated Theorem of Desargues.

Theorem 1.1. Let A₁A₂A₃ and B₁B₂B₃ be two triangles in such a position that the lines A_iB_i pass through a point O. Consider the points AiAj∩BiBj=Ck, where {i, j, k} = {1, 2, 3}. Then the points C₁, C₂, C₃ are on a line t.

Less formally, when the two triangles are in perspective from the point O then they are also in perspective from the line t. More details on Desargues’ theorem can be found in Coxeter’s book [48], where similar theorems, for example the Theorem of Pappus, are also discussed. These theorems will also occur in our book, mainly in the context of finite planes and spaces. In Chapters 2 and 3 we shall see how particular cases of Desargues’ theorem are related to properties of the coordinate structure of the projective plane. We shall also call the configuration of the ten points (A’s, B’s and C’s and O) and the ten lines (the lines A_iB_i, the sides of the two triangles and the line t) a closed Desargues configura...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Contents
• Preface
• 1. Definition of projective planes, examples
• 2. Basic properties of collineations and the Theorem of Baer
• 3. Coordinatization of projective planes
• 4. Projective spaces of higher dimensions
• 5. Higher dimensional representations
• 6. Arcs, ovals and blocking sets
• 7. (k; n)-arcs and multiple blocking sets
• 8. Algebraic curves and finite geometry
• 9. Arcs, caps, unitals and blocking sets in higher dimensional spaces
• 10. Generalized polygons, Möbius planes
• 11. Hyperovals
• 12. Some applications of finite geometry in combinatorics
• 13. Some applications of finite geometry in coding theory and cryptography
• Bibliography
• Index