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Group Theory

Dover Publications
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It is impossible not to be a fan of Dover Books. I was fortunate enough to discover this source of literary wealth in my teens and my enthusiasm has only grown over the years. The book under review is a wonderful example of why this is so: today’s textbooks’ price tags come in at roughly an order of magnitude more than what used to be the case when we (i.e. senior faculty) were students, and wouldn’t it be nice if we could use the books we were brought up with? (I admit to a creeping curmudgeonliness in my make-up, but that doesn’t imply I’m wrong!) Well, Dover is a source of grand old books — still more than able to do the job — at very low prices: in fact, prices even a student could afford!

The book under review, W. R. Scott’s Group Theory, is a fantastic introduction to the title’s subject, starting with the most foundational material, covering what I would characterize as a very strong undergraduate level course in the first six chapters, and then going well beyond this: the remaining nine chapters can easily be partitioned and parsed so as to cover two, or maybe three, advanced undergraduate seminars or graduate courses. With some guidance, Scott’s book can also serve well to prepare the student facing quals for this particular part of the graduate ordeal (or rite of passage).

In other words, Scott’s Group Theory has pretty much everything: a gorgeous treatment of Remak-Krull-Schmidt in Chapter 4, Sylow done with gusto at the start of Chapter 6 (which suggests that the indicated undergraduate course should be pitched pretty hard: pick your audience), and then on to such things as supersolvability, free groups, group extensions, permutation groups, symmetric and alternating groups, group representations, products of subgroups, and then some Wedderburn theory and material on infinite abelian groups. This is a pretty broad palette, to be sure, and requires a serious commitment on the part of the reader.

But the book is very well-written, if in the older non-chatty style: Scott is quite terse. I find this appealing, and, as far as group theory goes, preferable: the subject is fundamental, after all, and is therefore more autonomous than its derivative subjects. And the theory of finite groups is possibly the best example, this side of formal logic, of a self-contained sub-discipline in Mathematics.

There are over 500 exercises “in varying degrees of difficulty,” Scott is scrupulous about his proofs, and each chapter is capped off by a short list of references (dated, since this is the verbatim republication of the 1964 edition, but still very relevant).

Finally, a minor caveat. Unlike current texts, Scott’s Group Theory is nigh on void of motivational examples, notes to the reader, and so on: as I said, it’s the old non-chatty approach to Mathematics. If anything needs to be added, well, that’s what lectures are for. The style Scott favors is only a little more relaxed than Landau’s notorious telegraph style (which, by the way, I also like a great deal: “Just the facts, ma’am — and their proofs…”). It is especially conducive to the hard pencil and paper labor any would-be mathematician needs to do while reading and struggling with such serious material: for such a player this book is ideal!

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

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Tuesday, February 1, 2005
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Wednesday, June 29, 2011