INTRODUCTION TO FINITE GEOMETRIES by F. KÄRTESZI PROFESSOR AT THE EÖTVÖS LORAND UNIVERSITY BUDAPEST 1976 NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM · OXFORD AMERICAN ELSEVIER PUBLISHING COMPANY, INC. NEW YORK This monograph was published in Hungary as Vol. 7 in the series of DlSQUISmONES MATHEMATICAE HUNGARICAE Translated by L. Vekerdi English translation supervised by Minerva Translations Ltd., 5 Burntwood Grange Road, London S.W. 18 3JY Library of Congress Catalog Card Number: 74—83729 North-Holland ISBN: 0 7204 2832 7 American Elsevier ISBN: 0 444 10855 6 © Akadémiai Kiado · Budapest 1976 All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the Copyright owner. Joint edition published by North-Holland Publishing Company, Amsterdam—Oxford and Akadémiai Kiado, Publishing House of the Hungarian Academy of Sciences, Budapest Sole distributors for the U.S.A. and Canada: American Else vier Publishing Company, Inc., 52, Vanderbilt Avenue, New York, N.Y. 10017 Printed in Hungary To my dear Master, Professor Beniamino Segre PREFACE This book contains the material elaborated during my courses held at the Eötvös Lorând University of Budapest under the title "projective geometry" since 1948. In the beginning I briefly mentioned the notation of finite projective planes in connection with the classical projective geometry. Over the years I continuously enlarged the share of finite geometries at the expense of the classical themes. As the preliminary training of my students varied widely, I had to begin at a quite elementary level. More than two decades experience in teaching as well as the rapid development of the subject over the last three decades gave me a didactical point of view which was mirrored in my lectures. Encouraged by some former students of mine I have presented these lectures in the form of an introductory textbook of a didactical character. As to the treatment, my book is presented in a rather unusual form in com- parison with traditional and modern textbooks alike. First of all, I do not strive to give a complete account of the subject up to the present day but rather the ways and means which led to its development. I consider this book to be somewhat ex- perimental — as indeed my lectures were — and I await the opinion of the readers who begin to learn about the theory of finite geometries through the pages of this book. As to the terminology — mainly because of the shortage of relevant publica- tions in Hungarian — I was often obliged to introduce new notations or to deviate from the usual ones; of course in such cases I have always explained the exact meaning of the terms. Similarly, certain illustrations, such as tables, deformed figures, etc., are presented in an unusual form. I should like to express my indebtedness to Professors Gy. Strommer and G. Szâsz for their useful advice and careful work in supervising the manuscript of this book, to M. Frigyesi and G. Vidéki for the clarity and neatness of the figures. Moreover, I owe thanks to the Akadémiai Kiado (Publishing House of the Hungarian Academy of Sciences) for the publication of the book. F. Karteszi COMMON NOTATION |H| number of the elements of a set H. {a,b,..... .*> the set consisting of the elements a,b,...,h. {...\...} the set of the elements to the left of the dividing line which all have the property denoted on the right hand side of the dividing line. e.g. {x£G|x2=l}. xeH the element x belongs to the set H. x$H x does not belong to H. UCH every element of set U belongs to H. UcH every element of U belongs to H but not every element of H belongs to U. AnB {χ\χζ A and xÇB}, greatest common subset, intersection. AuB {x\x£A or xÇB}, union of A and B. A®B the set of all ordered pairs (a, b) where αζ A and è£B. A-Bor A\B If BgA, {x|*€A and x$B}. o, *, 1. ,λ signs of operations (with meaning to be specified from case to case). (A, 0) structure with one operation : written (A, +) or (A, x) when the operation is addition or multiplication, respec- tively. (A, +, ·), (A,_L, A) structures with two operations, Fixmb) ternary operation. GF{q) Galois field of q elements, (x x , Xs) lt 2 point-coordinates, [«i.Ma.Ma] line-coordinates. • sign of incidence (on an incidence table), 2 (arbitrary) incidence table, Ω cyclic incidence table, Qm,x parcel (on an incidence table), r(q\r the parcelled incidence table of order q. AB; a,... ,x vectors, Ax vector multiplied by a scalar. COMMON NOTATION Xlll (x) and [u] point and line determined by their coordinates. [u] (x) t/i*i + ux + ux. 2 2 z 3 S the «-dimensional (arithmetic) space over the coordinate n>q field GF(q) ; Galois space. t-(v, k, λ) the t-(v, k, λ) block system. I the axiom system consisting of the axioms l l , l , l . 1? 2 3 4 D—, P—, R—, Desarguesian, Pascalian, Reidemeister. MD, MP, micro-Desarguesian, micro-Pascalian, MR micro-Reidemeister. CHAPTER 1 BASIC CONCEPTS CONCERNING FINITE GEOMETRIES In this chapter a simple and elementary treatment will be given to some basic concepts, problems and methods of finite geometries. To begin with we shall out- line what this new branch of mathematics is about and how it developed out of several classical subjects. 1.1 The finite plane The projective plane obtained by extending the Euclidean plane by ideal points and by the ideal line is called the classical projective plane. On this plane the fol- lowing basic facts concerning the incidence relation of points and lines are known: (1) Given any two distinct points there exists just one line incident with both of them (called the connecting line of the two points). (2) Given any two distinct lines there exists just one point incident with both of them (called the point of intersection of the two lines). (3) There exist four points such that a line incident with any two of them is not incident with either of the remaining two. (We shall express this by saying that there exists a real quadrangle.) Let us now consider the basic facts enumerated above as axioms. Every theorem which is a consequence of these axioms is also a theorem on the classical projective plane. There are, however, other systems consisting of objects called points and objects called lines together with a relation called incidence — which either does or does not hold for any pair consisting of one point and one line — and satisfying the requirements of (1), (2), and (3). The oldest and most simple example of this is the Fano figure. (See Fig. 2.) Let us consider an uncoloured chessboard of 7 squares by 7 squares and upon it 21 chessmen in the arrangement that can be seen in Fig. 1. Let the columns of the chessboard be called "points" P P , P , P , P , P , and P and let the l9 2 3 4 5 6 7 rows be called "lines" l / , / , / , / , / and / . Let the "incidence" of a point and l9 2 3 4 5 e 7 a line be interpreted as the fact that the square at the intersection of the corre- sponding row and the column is occupied by a chessman — i.e. on the figure by a sign o — and let an empty square be interpreted as "non-incidence". It is easy 2 BASIC CONCEPTS to see that this model in fact fulfils the requirements of (1), (2) and (3). — Thus, for instance, the only line connecting points P and P is / . The only point inci- 4 6 e dent with both of the lines / and / is P . A real quadrangle is given by P P P P^ 3 5 7 X 2 Z the six lines connecting pairs of points of this quadrangle are l / , / , / , /«$ and / l9 2 3 5 7 so the system satisfies (3). In what follows, this system will be called a Fano plane. R p2 p3 p. p5 Ps p7 •1 li • • h • • • l3 • • • It • • • Is • • • u • • • l7 • • m Figure 1 We shall principally be concerned with the investigation of finite planes; the treatment of a plane as a point set has many advantages. Thus the line appears as a subset defined by certain properties. So the plane is a point set, say, Σ={Ρ Ρ ,...} and the lines of the plane are certain subsets of this set Σ. We 19 2 shall denote points by upper case Latin letters, lines by lower case Latin letters. The projective plane will be defined by the following axiom system: l Ι/ΡζΣ and Ο,ζΣ and P^Q, then there exists uniquely a line I for which ΡζΙ x andQtl. '2 ΙίΖ^Σ and lei andg^l, then there exists P for which P£g and ΡζΙ. I There exist four points determining two by two six distinct lines according 3 to ! x We know already that the abstract projective plane defined by these three axioms is not an empty concept; we have the examples of the classical projective plane and the Fano plane. (a) It follows already from axioms l and l that any two distinct lines have one x 2 and only one common point. Namely according to l there exists one and according to l there cannot exist 2 x more than one common point of two distinct lines. THE FINITE PLANE 3 (b) There exist four lines of which no three have a common point. (That is, there exist real quadrilaterals.) Namely, if P P , P and P are four points satisfying axiom l, then the lines l9 2 3 4 3 P P , P P*, PzP* and Λ^ι correspond to the statement (f>). 1 2 2 The Principle of Duality, as in classical projective geometry, is valid on the abstract projective plane as well. This follows from the fact that l and l and, t 2 furthermore, l and Theorem (f>) are the duals of each other. 3 (c) Any line contains at least three points; each point is contained in at least three lines. The second part of the statement is the dual of the first, therefore it suffices to prove the first part. There exist real quadrangles; let one of them be the quadrangle ΡχΡ^Ρ^Ρ^ An arbitrary line / of the plane may contain either two, one or none of the vertices of its quadrangle. In the first case, suppose P £/ and P €/. The lines P P and 3 4 X 2 P P have a unique common point which is distinct from the vertices of the quad- Z A rangle. This point, however, is already a third point of line /. In the second case suppose / only passes through the vertex P then the lines Ρ Ρ , P P each meet 4 Χ 2 2 3 / in one point. In the third case the lines Ρ Ρχ, P P and P P each meet / in one 4 4 2 4 3 point. Thus our statement (c) is true. These theorems, which reflect the basic properties of the classical projective plane, define, if taken as axioms, a much too general notion of the abstract projective plane. We introduce a fourth axiom in order to restrict this concept and obtain the notion of the finite projective plane of order q. I There exists a line which consists ofq + l points, where q(>l)is a (suitable) 4 positive integer. In fact, this axiom can be considered a reasonable one, since it is satisfied by the Fano plane namely with q=2. Furthermore, it follows from the former three axioms that #>1. (Theorem (c).) The axiom system consisting of l 1 , l and l will be denoted by I. We shall l5 2 3 4 now discuss some simple but significant consequences of the axiom system I. (à) Every line consists ofq + l points. (e) There are q + 1 lines through every point. (f) The plane consists ofq2+q + l points. (g) The plane contains q2 + q +1 lines. In the example given in Fig. 1, where q=2, these theorems are exhibited as the following properties: 2+1=3 signs occur in each row of the table. 3 signs occur in each column of the table. 2 Introduction 4 BASIC CONCEPTS The table consists of 22+2 + l=7 (non-empty) columns. The table consists of 7 (non-empty) rows. And now we shall prove statements (d) to (g) successively. The perspective correlation of ranges and pencils on the classical projective plane is a notion which can also be extended to the plane defined by the axiom system I and the proof of our statements is based upon this one-to-one perspective correspondence. (d) Let l={P P , ...,P } be a line satisfying l and let /' be an arbitrary l9 2 q+1 4 line distinct from /, i.e. /' ^ /. According to l , the lines /and /' have a common point ; 2 let their common point be P.. Line V contains — according to Theorem (c) — a point P distinct from Pj\ furthermore, the line connecting the point P of /, dis- r k tinct from Pj, with P also contains a point P which is distinct from both P and r s r P . Now the lines connecting the point P with the points of the line / exhaust k s the totality of the lines passing through the point P — as can be seen from axioms s l and l — and each of them meets /' in a point; thus the line V also contains q+1 x 2 points. Clearly, V cannot contain more than q+\ points because otherwise, by connecting them with the point P, we would obtain more than q+l lines in the s pencil through P. s (e) By dualizing the proof of Theorem (d) we obtain the proof of Theorem (e), since we know already that the number of lines passing through point P s is #+1. (f) Every point of the plane is, according to \ a point on a line of the pencil l9 through P. On each of the q+1 lines of this pencil — according to (d) — there s are q points, distinct from P. Thus the total number of the points is: (q-\- l)q+1 = s = q*+q+l. (g) The proof of Theorem (g) is given by dualizing the proof of Theorem (f). Definition: The figure corresponding to the axiom system I is said to be a finite projective * 'plane of order q". Where no misunderstanding can occur we shall abbreviate this to "plane of order q". The following question remains to be answered: Does a plane of order q exist if q is an integer greater than 2? 1.2 Isomorphic planes, incidence tables Let us consider two finite projective planes of order q i.e. two sets having the 9 same number, q2+q+1 = «, of elements in which there can be found n subsets each having q+l elements so that the structure of the two sets generated by these sub- sets — we shall call it the combinatorial structure — should satisfy the axiom sys- tem I. As both sets have the same number of elements, the same names, i.e. in- dices, can be used to denote the elements in each set.