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Algebra and Geometry

Alan F. Beardon
Cambridge University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Stephen Ahearn
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Alan Beardon's Algebra and Geometry introduces the concepts of algebra, primarily group theory and linear algebra, by focusing on classical geometric maps: symmetries, isometries, linear transformations, and Möbius transformations. Beardon's goal is to present some beautiful material in a way that emphasizes the unity of mathematics over the compartmentalization often found in mathematics curricula, and at a level that is accessible to a junior or senior mathematics major. It is a vision for a beautiful book but, unfortunately, he only partially succeeds in fulfilling it.

Naturally, the book does not cover all of the topics typically found in an undergraduate algebra or geometry text. Beardon focuses on groups and vector spaces. The basics of these subjects are introduced and then given meaning through geometry. For example, eigenvalues and orthogonal matrices are introduced so that the isometries of Euclidean space can be described; the group of Möbius transformations is shown to be isomorphic to the quotient of the special linear group (with complex entries) and the subgroup {±I}. Spherical geometry and Euler's formula for a sphere are introduced so that the Platonic solids can be identified.

The algebraic ideas are tied together nicely by the geometric theme. The algebraic concepts are all introduced for the purpose of describing or studying geometric maps: symmetries, isometries, etc. However, I see the book in this light only in hindsight. This focus did not become clear to me until near the end of the book and I do not believe that most students will see this common thread.

The material is accessible to upper-level undergraduate mathematics majors. The pace of the book is good: low dimensional cases are presented before the general n-dimensional case is presented; groups are introduced long before their properties are discussed. The exercises are a nice mix of concrete calculations and abstract proofs. The book, however, could have been more accessible to students. For a book with geometry in the title, there are very few pictures. The exposition would be greatly enhanced if there were more diagrams to go along with the written descriptions.

Furthermore, in some sections, particularly early in the book, there are decidedly few examples. For example, in the section in which groups are introduced, the only example is the group of linear functions on the reals under the operation of composition. Moreover, when examples are given, they tend to come after the theorems and their proofs. The example of the group of linear functions, for example, comes at the end of the section. The exercises do make up somewhat for the lack of examples, as they provide a number of concrete examples and calculations but, this is not the order for best learning the material. The examples should come first and then the theorems.

So I long for what might have been. Beardon has written a fine introductory text. It could have been great. Despite my longing for what might have been, let me encourage you to consider using Algebra and Geometry, if you do not want to teach a standard abstract algebra course. Beardon outlines a beautiful course.

Stephen T. Ahearn ( teaches mathematics at Macalester College in St. Paul, MN. His primary research interests are in algebraic topology and computational topology/geometry but allows himself to be distracted by other interesting topics as in his article "Tolstoy's Integration Metaphor from War and Peace." He also enjoys hiking, swimming, baking bread, and reading.

 1. Groups and permutations; 2. The real numbers; 3. The complex plane; 4. Vectors in three-dimensional space; 5. Spherical geometry; 6. Quaternions and isometries; 7. Vector spaces; 8. Linear equations; 9. Matrices; 10. Eigenvectors; 11. Orthonormal bases of Euclidean space; 12. Groups; 13. The Möbius group; 14. Group actions; 15. Hyperbolic geometry.