The title of the book is somewhat misleading. This is not a book that one would want to use for a course such as Introduction to Abstract Algebra. Rather, as it is explained in the preface, it is a textbook meant for a "bridge" course, such as Transition to Higher Mathematics, that is, for a course that teaches the concept and technique of proofs. Such courses always come before the student would take a course in abstract algebra course, and are, in most departments, required for all mathematics majors.

For that purpose, this is certainly an ambitious book. The content of the first two chapters includes sets, functions, relations, binary operations, integers, and mathematical induction, would probably be included in every such course, together with some electives such as matrices and permutations. The latter are also justified since the book will later focus on abstract algebra.

Then we get to the difficult part of the book. The four middle chapters discuss groups, rings, and fields. This is very courageous to do in this early class. This reviewer has taught Transition to Higher Mathematics several times at a decent university, and his students had enough problems learning the concept and techniques of proofs even without the added difficulty of the highly abstract concepts covered here. Furthermore, students of Abstract Algebra, who have already take a Transition course, still have enough problems learning the abstract concepts that dominate Group theory, Ring theory, and Field theory.

Attempting to join a Transition course and an Abstract Algebra course into one course may be too much. This reviewer agrees that if we are to teach serious proof techniques, we need serious material in which to illustrate those techniques. One could debate whether abstract algebra is a good choice for that material since it is indeed more abstract than anything the students have seen before. A more hands-on topic such as combinatorial enumeration or the theory of algorithms could provide a gentler introduction to higher mathematics.

Even if we are to choose abstract algebra, however, concepts like the normalizer, normal subgroups and quotient groups may be too much for this class. These should be covered in a regular Abstract Algebra course.

The book ends with two chapters, Complex Numbers, and Polynomials, which would again be on every instructor's list, just as the first two chapters. It would perhaps be better to discuss these topics earlier in the text, and then use them as examples later.

The number and level of exercises, and the style of writing is appropriate. If you are looking for a book combining the aforementioned topics, then you should check this book out, since there are not many books that do that.

Miklós Bóna is Associate Professor of Mathematics at the University of Florida.