Visual Group Theory

Description

This text approaches the learning of group theory visually. It allows the student to see groups, experiment with groups and understand their significance. It brings groups, subgroups, homomorphisms, products, and quotients into clear view. Every topic and theorem is accompanied with a visual demonstration of its meaning and import, from the basics of groups and subgroups through advanced structural concepts such as semidirect products and Sylow theory. Opening chapters anchor the reader's intuitions with puzzles and symmetrical objects, defining groups as collections of actions. This approach gives early access to Cayley diagrams, the visualization technique central to the book, due to its unique ability to make group structure visually evident. This book is ideal as a supplement for a first course in group theory or alternatively as recreational reading.

The author's intentions are clear. In the preface, he writes, "Most textbooks present the theory of groups using theorems, proofs, and examples. Their exercises teach you to make conjectures about groups and prove or refute them. This book, however, teaches you to know groups.'' Using hundreds of illustrations and examples from art, architecture, dancing and chemistry, the author takes a fresh look at the presentation of group theory. The result is an approach that stresses exploration using Cayley diagrams and thinking of groups as collections of actions rather than sets with an endowed binary operation. It is not clear that students will appreciate this approach or be better off for it, but the book makes a convincing argument for trying.
The highlight of the book is meant to be the illustrations: there are over 300 of them. They rarely feel superfluous and oftentimes do their job of making the reader say, "I see what is going on here.'' The Cayley diagram is the most common illustration in the book and helps the reader visualize the structure of the group. In addition, the author extends these diagrams to explain notions such as cosets and semidirect products. As these are two notions that often cause considerable confusion amongst undergraduates, these pictures should be an excellent supplemental resource for both the student and the instructor.
The book covers most of the usual topics of an introductory group theory text, but certainly favors some topics more than others. For instance, direct and semidirect products occur earlier and more often. However, topics such as the center of a group, centralizers and matrix groups are ignored almost completely.
There are several errors in the book, mostly of the benign typographical variety. The author maintains an errata website (http://web.bentley.edu/empl/c/ncarter/vgt/errata.html) that seems to have caught the most glaring issues. Also available on the author's personal web page is a free software package called Group Explorer. This program is referred to several times in the book and for a student unfamiliar with computer algebra systems, this might be of some value.
A student considering graduate school in mathematics will probably feel under-prepared if this is his or her only exposure to group theory (the standard definition of a group does not come until Chapter 4 and even then misleads the reader into thinking a set can be a group), but if you are teaching group theory as part of a program leading away from more abstract mathematics (e.g., math education, actuarial sciences, engineering), the approach in this book seems far more likely to capture the interest of students. The exercises are tractable and, for the first time, many students of group theory will feel that they can tell a less mathematically inclined person what it is they study.

Reviewed by Adam Glesser