Introduction to finite geometries / by F. Kárteszi ; [translated by L. Vekerdi]

Bibliogr. p. 263-264. Index

This book is based on a course that has apparently been taught at the Hungarian undergraduate level. Its purpose is clearly didactic. Topics treated include finite projective, hyperbolic and affine planes, incidence tables, Latin squares, Galois spaces, conics in Galois planes, collineations, configurations, graphs related to Galois planes and to configurations, block designs, etc. The methods are mainly based on combinatorial notions and on coordinate systems, with some group theory here and there. The author adopts a pace that seems well adapted to the possibilities of students, and there are 70 good problems of various levels of difficulty.
The reviewer has some difficulty in distinguishing the unifying ideas of this work. To take but one example, conics are studied at length with a proof of Segre's important theorem on ovals in odd order Galois planes. However, Pascal's theorem is not even mentioned, while the Pappus-Pascal theorem for two intersecting lines appears at several places. In addition, Segre's theorem is so important because of its applications to higher dimensional quadrics and because it leads to several new problems. There is no discussion of these topics, despite a short section on inversive planes and ovoids.
The author claims to aim at no complete account of the subject but rather at the ways and means leading to its development. On this basis it is surprising to find no mention of any open problems, not even the case of the plane of order 10. It is still more surprising to find so little historical data, as for instance in the discussion of Latin squares, where the roles of Euler and Tarry are pointed out but where no mention is made of the spectacular Bruck-Ryser theorem, these names appearing however in the short bibliography. In the same order of ideas the author observes that the subject is developing rapidly, but his most recent reference is to the year 1969. The Hungarian original was published in 1972.
A student using this book should be prepared to encounter a quite confusing situation. On p. 36 he will first meet Pascal's theorem without any definition; the same theorem appears again on p. 162 as the Pappus-Pascal theorem, and its precise statement is finally given in Problem 17 at the end of the section but without any reference to Pappus or to Pascal. The short index is of no help in this and in similar situations. (MathSciNet)