I highlight here three essential aspects of this book’s nonstandard approach to group theory, and briefly discuss its organization.

First and foremost, images and visual examples are the heart of this book. There are more than 300 images, an average of more than one per page. The most used visualization tool is Cayley diagrams (defined in Chapter 2) because they represent group structure clearly and faithfully. But multiplication tables and objects with symmetry also appear regularly, and to a lesser extent cycle graphs, Hasse diagrams, action diagrams, homomorphism diagrams, and more. As you can tell by flipping through the pages,...

In 1974, Ernö Rubik of Budapest, Hungary unleashed his fascinating invention called Rubik’s Cube. It infiltrated popular culture, appearing in feature films, inspiring competitions, and captivating children and geniuses alike. Mathematics journals carried research articles analyzing the cube and its patterns. Those unable to solve the cube could learn solutions from any of dozens of books.

A new Rubik’s Cube comes out of the box looking like Figure 1.1. Each face of the cube contains nine smaller faces of smaller cubes, with the colors arranged to agree. You begin playing with the cube by rotating its faces to mix up...

Chapter 1 introduced group theory by examining those properties of Rubik’s Cube that make it attractive to beginners. Let us now investigate those aspects of the cube that make it difficult to solve.

When you have in your hands an unsolved Rubik’s Cube, and you do not know a method for solving it, experimentally twisting the faces quickly begins to seem pointless. You are wandering aimlessly through the multitude of possible configurations of the cube. Somewhere in this enormous wilderness of jumbled cube configurations is the one oasis you want to find—the solved cube. But without any idea of...

The groups you’ve seen in this book so far may leave you wondering what the purpose of group theory is. After all, the things we’ve studied have little value other than intellectual amusement—games, puzzles, and imaginary situations like coins on a table. If group theory were just a collection of intellectual amusements, anyone could be forgiven for asking, “So what?”

The two chapters we’ve spent learning the basics have positioned us to learn some of group theory’s more practical applications. This chapter tours a few of those applications, and provides references to where interested readers can go to explore...

So far you have been learning about groups in a way that is unique to this book. Our unofficial definition of a group, Definition 1.9, is not how mathematicians define a group. Though there are many benefits to the approach I have taken (and will continue using later), it would be an incomplete education in group theory that did not acquaint you with the official definition of a group, the one you would find in anyothergroup theory book.

That standard definition comes with its own natural visualization technique, called a multiplication table. Since we are focusing on visualization,...

We have learned two powerful group theory visualization techniques. Cayley diagrams show us groups as collections of actions, and multiplication tables show them to us as binary operations. This chapter uses both to give a well-rounded introduction to five famous families of groups. Along the way a variety of new concepts will also arise.

We will begin by meeting the cyclic groups, for many reasons the perfect place to start our tour. Not only are cyclic groups the simplest kind of symmetry groups, but meeting them first will make the rest of the chapter clearer: Cyclic groups show up in...

Though the chapter title may not suggest it, you’ve come to the exciting part of this book! Through five chapters, you’ve gained a lot of familiarity, learned the lay of the group theory land, and are now ready for in-depth study. Starting with this chapter, we’ll be getting more analytical and learning more mathematical terminology,but without giving up our visual roots. Entering advanced realms doesn’t mean leaving visualization behind; it can be just as helpful in advanced areas as in introductory ones, and often moreso. In fact, the degree to which visualization has helped me better understand the material...

Chapter 6 looked inside groups to find subgroups, and so taught us something about the groups’ internal structure. This chapter moves in the opposite direction, showing how groups can be assembled together to construct larger groups. We will learn two such construction processes, each a different kind of group product operation. Here I’m using the word “product” in its mathematical sense, meaning a kind of multiplication. We will also learn how certain subgroups can be the key to reversing such a process, facilitating a quotient operation that deconstructs any kind of product, revealing how a larger group may be constructed...

Throughout this book I’ve said things like “this group has the same structure as that group” or “there is a copy of this group inside that group.” The first time I did so (page 19, regarding the equivalence of Figures 2.7 and 2.8), I carefully explained how the two structures were the same. But since then, when stating that two structures are the same, I’ve depended on your ability to see how their patterns match, without my spelling out all the details. The purpose of this chapter is to create and study precise descriptions of how two structures correspond, because...

This chapter is about one question: What groups are there? In Chapter 5, we met a variety of groups, to gain a breadth of exposure to the subject. But it was just a sample, not a comprehensive list. We have since seen groups outside the five families of that chapter. Seeing a list of all groups would surely benefit our knowledge of (and intuition for) group theory. This chapter begins to make such a list.

Of course, there are infinitely many groups, so we cannot list them all! But we can start with the smallest ones and grow our list...

This book ends where group theory began. In the nineteenth century, two very young mathematicians, Neils Abel and Évariste Galois, answered a question about which the mathematical world had been curious for centuries. It is a question about the central activity of algebra, solving equations. Galois coined a new term to describe the mathematical objects that played a central role in his work on this question; he called them groups, and group theory was born.

But groups are only part of the solution; the work of Abel and Galois also involves algebraic structures calledfields. I touch on field theory...