★★★★★ The abstraction of abstraction

Abstract Algebra

David S. Dummit, Richard M. Foote

Most people think they know what algebra is. We take a course in high school called "Algebra" in which we learn the clever trick of using a letter to represent a number. For instance, suppose you know that there's an 8% sales tax on every Christmas present you may buy. If you have $50, how expensive a gift can you buy? Here's the trick: use p for the list price. Then with tax, it's going to cost you p+0.08×p, because 0.08×p is 8% of p. But p+0.08×p = 1×p+0.08×p = 1.08×p. If we set that to $50, i.e.

1.08×p = 50,

then we can divide by 1.08 to get p = $46.29. That's the most expensive gift you can buy with $50. So, that's nice, but it seems like a lot of trouble to solve a simple problem.

But here's the thing, you haven't just solved this problem. You've solved the problem of finding out how much you can buy with $n, no matter what n is: it's always going to be n/1.08. In fact, if the sales tax is s, then the most you can buy is n/(1+s). That's the big benefit of algebra: you can solve not just one problem, but every problem of this kind once and for all. It was the great Persian mathematician Muhammad ibn Musa al-Khwarizmi who had this brilliant idea. By solving an abstract version of the problem, you solve not just this problem, but every problem of this type.

So, what is ABSTRACT algebra? Mathematicians like to turn things on themselves. Suppose we ask, what kinds of problem can we solve by abstraction? Let's try to solve that once and for all. We'll use symbols to represent not just numbers, but other things. What kind of other things? We're not going to tell you. We're only going to ask that the things obey certain rules (which we call axioms). So, for instance, the three group axioms are

1. (a★b)★c = a★(b★c).
2. 1★a = a★1 = a.
3. a^-1★a = a★a^-1 = 1.

These, somewhat simplified, are the axioms of Group Theory. You will find them on p 17 of David S. Dummit and Richard M. Foote's Abstract Algebra. This looks like a very thin basis on which to start, but these three axioms are enough to form the foundation of an astonishingly rich theory with applications to all of mathematics and physics. The first six chapters of Dummit and Foote are concerned with Group Theory. These are followed by three chapters on Ring Theory. A ring is more complicated than a group, but it is composed of two connected groups, so with Group Theory under your belt (somewhat), you're ready to tackle Ring Theory. And so on.

Now, this is where I have to confess that I have not literally read Dummit and Foote from cover to cover. I self-studied chapters 0-5 and most of 6 (thus, the Group Theory chapters), working all the exercises. Then I took a second semester course in Abstract Algebra, in which we covered material in chapters 7 - 14 of Dummit and Foote. Chapters 15 - 19 I have never studied.

I want to highlight the phrase "working all the exercises". The astonishing thing about Dummit and Foote is the hundreds (literally) of exercises. For instance, Chapter 1, "Introduction to Groups" has seven sections, with 36, 18, 20, 11, 3, 26, and 23 exercises. Many of the exercises have multiple parts. The easiest are very easy, but the most difficult are virtually self-contained research problems. No solutions are provided, but an Internet search will turn up collections of solutions.

To highlight what the exercises are like, I include the following picture, which I made after working Exercise 7 of section 5.5. In it the student is asked to characterize the thirteen groups of size 56. (That is, every group of size 56 is isomorphic to one of these thirteen.) I made the picture by generating a cycle graph of each group. If you look closely at the picture you will see that there are thirteen graphs in it, each containing 56 red dots connected by green lines. These are all the ways of arranging 56 things so that they satisfy the Group Theory axioms.

Dummit and Foote is an extraordinary resource for learning Abstract Algebra. Because of the hundreds of advanced exercises, it especially rewards self-study, if you're willing to work!

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