This is a remarkable text designed for highly motivated undergraduates having some preparation in linear algebra and some other post-calculus mathematics. It is noteworthy for its contents and the style of presentation. In the preface, the author lists three principles that he followed (briefly: examples should motivate definitions, technical points are presented only if needed later in the book, topics should be important for the average mathematician) and takes pains to point out that ``Do it the way you were taught'' is not one of them. The style throughout the text is to present basic concepts, give many nontrivial examples and present brief and understandable discussions of advanced material. The author has blended an introduction to groups, vector spaces and linear transformations to give an elegant chapter (5, Symmetry) giving a thorough discussion of Motions of the plane, Finite and discrete groups of motions and a classification of the Finite subgroups of the rotation group in three dimensions. As an example of the scope of the treatment, consider the 40-page Chapter 6, More on group theory. It begins with the standard material on the class equation, center of a p-group, Sylow's theorem and symmetric group (with extended examples along the way) but then follows with sections on Free groups, Generators and relations, and the Todd-Coxeter algorithm. Enough material is presented to give the student a good idea of what the topics are about. In keeping with the author's stated preference for geometric topics, the theory of bilinear forms is presented in some detail and then followed by chapters on linear groups and group representations. After presenting the expected topics through the orthogonality relations for irreducible characters of finite groups, there follows a brief discussion of continuous representations of compact groups. The representations of the circle group U1 are found and then used to discuss the representations of SO2. The later chapters (10–14) treat Rings, Factorization, Modules, Fields and Galois theory. Each of these chapters contains some special topics that one might not expect to see in a more standard text. For example, the chapter on factorization contains some detailed specific and general computations with quadratic fields and ideal class groups. The chapter on fields contains mention of Riemann surfaces. Each chapter is followed by many exercises organized into subsets corresponding to topic headings from the chapter. Many of these are \vadjust\vskip36pt% quite challenging, most are quite interesting. The student who works through this book will gain knowledge of the topics in standard algebra and learn of the connections with much classical and advanced mathematics. The expert who reads this book will appreciate having in one source many connections between the various mathematical specialties.
Reviewer: Janusz, Gerald J. [form MathSciNet]