Finite Geometries

This text serves as a nice introduction to the study of finite geometries. The first several chapters discuss projective planes. The authors treat higher dimensional projective spaces in a separate chapter, which allows a reader to focus only on dimension two, if desired. The next few chapters introduce particular collections of points in the form of arcs and blocking sets. Again, the consideration of these objects in higher dimensional spaces is given in a separate chapter. A further chapter discusses \( (q+2) \)-arcs in a projective plane with \( q \) points. Two chapters are given to generalizing projective planes in the form of generalized polygons, and discussing Mobius planes, where every three points determine a circle. The final two chapters are devoted to applications to combinatorics and coding theory. Each chapter contains some exercises, about 10 on average.

 

As the authors claim, the background necessary to engage successfully with this book is on the minimal side.  It does require some understanding of elementary group theory (mostly symmetric groups), finite fields, and linear algebra over finite fields. The book does not specifically require any knowledge of classical geometry, but neither is that developed. The classical projective and affine planes are mentioned, but mostly in passing. 

 

This book has a nice selection of material and is suitable for a wide range of readers, from advanced undergraduates to graduate students. It is an introduction to the field, but goes beyond what would be expected from a first course. For example, it contains a fair number of recent results. Overall, the book is well done, and helpfully, it includes a rich bibliography.

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Michele Intermont is an Associate Professor of Mathematics at Kalamazoo College. Her main interest is in algebraic topology.