Most of the mathematicians I know already have an opinion on the question “What is the best book from which to learn abstract algebra?” Maybe it is the book they used as undergraduates, or maybe it is a book they have taught from. Maybe they prefer the more traditional groups-first approach or maybe they prefer the rings-first approach. In any event, there are many books on the market that one could choose from, with a large amount of variation in the topics they cover and the approaches they use. It therefore seems like it would be hard for a new book to come out with a new viewpoint on the field. However, Nathan Carter’s *Visual Group Theory*, published this summer by the MAA, has done just that.

To be precise, Carter’s book is not a full abstract algebra book but rather a book about group theory with a few topics about fields covered at the end. In the preface, Carter emphasizes that he calls it a book “*about* group theory because this book does not aim to cover group theory comprehensively.” Given that disclaimer, I was surprised to find that most of the topics I would cover in a first semester of group theory are indeed covered in this book: subgroups, Lagrange’s Theorem, products and quotients, conjugacy, homomorphisms, the Fundamental Theorem of Abelian Groups, Sylow’s Theorems, and Galois theory. But before Carter gets to any of that, he has five full chapters dedicated to defining, motivating, and giving examples of groups, and this is where the difference in his approach shows through most clearly.

In most group theory books, one defines a group as a binary operation satisfying certain properties, and then proves the structure that follows from these properties, eventually proving Cayley’s Theorem which says that every group is actually a subgroup of some permutation group. Carter turns this approach on its head, defining a group as a collection of permutations of some kind of object and eventually showing how this leads to a natural binary operation that one can define. One nice thing that comes out of this approach is that the reader gets to do everything very concretely and visually, as Carter puts Cayley diagrams at the heart of much of the discussion.

In addition to Cayley diagrams, Carter’s book lives up to its title as he uses pictures of Rubiks Cubes, molecules, and many other things in order to show various symmetries and discuss the structures that they can lead to. These color illustrations, many of which were done with the free computer program Group Explorer (which Carter helped create and recommends highly), are used throughout the book to explore many of the main ideas in group theory. For some of the topics I found the pictures to be unnecessary or even confusing, but in most of the book I felt that they would make students feel much better about the material.

There is no doubt that Carter’s book contains a number of novel ideas and approaches to the material, and many of these would force an instructor to make an all-or-nothing decision rather than sample some of the approaches in bits or pieces while teaching primarily from another book. Many of his ideas have some appeal, but I wish that he had spent time (perhaps in an introduction) elaborating on why he thinks this approach is more valuable than other approaches. In particular, this reviewer walked away from the book unconvinced that I would be willing to dive in and use his book the next time I teach the course.

The book has a lot going for it, however — Carter’s exposition manages to find a nice balance of technical material and chattiness, and there are a large number of good exercises, including several designed to help the reader use the *Group Explorer *program. I was most impressed with the concluding chapter on Galois theory, in which Carter does a very nice job of cutting away the fat and getting right to the heart of Galois theory in a way that I think students could really appreciate in a short period of time. While I do not think it has yet won me over as a book to base a course on, I would not hesitate in recommending this book to a motivated student who wished to experience some group theory on their own.

Darren Glass is an Associate Professor of mathematics at Gettysburg College. His main mathematical interests include Galois theory, so perhaps this review betrays some of his own biases. Oh well. He can be reached at [email protected].