Read This!

The MAA Online book review column

Briefly Noted

July 2004

I once had a friend whose all-consuming passion was the theory of finite groups.  When Gorenstein's dream of the classification of the finite simple groups was achieved, my friend half-joked that a lot of mathematicians, himself included, had just been robbed of their raison d'être.  He made some noise about John Horton Conway's monstrous moonshine, which was music to my number-theoretic ears, of course, but, as far as I know, he didn't go anywhere with it.  I lost track over the years, both of my friend and of finite group theory.  Regarding the latter, however, I found out recently that cross-fertilization from physics has evidently opened up some very beautiful and dramatic lines, having to do with vertex algebras, originating in physics.  The original moonshine phenomenon, noted first by McKay and then expounded by Conway and Norton, is now part of a brightly lit landscape (with Borcherds' work figuring very prominantly), but apparently some related material is still hidden in dark shadows and a lot of exciting things may be expected.  (See Terry Gannon's fascinating recent arXiv article, "Monstrous Moonshine: the first twenty-five years," forthcoming in Bull. London Math. Soc.)  

So, indeed, the theory of finite groups is alive and kicking.  Moreover, it would be disingenuous to suggest that the field has been taken over by methodologies coming from physics and number theory.  The book under review is incontrovertible proof that the theory of finite groups per se is alive and well, too.  Indeed, while serving to introduce a relative novice to the subject, The Theory of Finite Groups: An Introduction is also a  marvelous treatment of a large chunk of what's going on today. It is presented as "the first book which shows us the amalgam method and moreover shows us how it works," this according to G. Stroth's Zentralblatt review, cited on the back cover.  The authors' discussion of this important method can be found on p. 281 ff.  They note that "[the amalgam] method was introduced by Goldschmidt at the end of the 1970's and since then has become an integral part of the local structure theory of finite groups.  The name amalgam method refers to the fact that this method does not require a finite group but can be carried out already in the amalgamated product of ... finite groups."  This is, in and of itself, very interesting material and obviously of interest to any aspiring hard-core group theorist. 

And, as already hinted, there's a lot more to this book.  It starts with baby group theory, so to speak, using the notion of a group action both systematically and often, very often: it's a central pedagogical and expositional device, which works well!  This introductory material is presented at a fast pace: Sylow appears on p. 62.  But it works, even if the reader really ought to be mathematically mature beyond the level of a garden-variety junior mathematics major. 

The book goes on to finite group theory properly so-called, as the philosophers say, while omitting representation-theoretic aspects as well as any real treatment of the classification problem.  (The only space devoted to the classification as such is a statement of the theorem on pp. 370-371: they authors are busy about other things.)  There are a lot of nice exercises, the scholarship is phenomenally thorough, and many very interesting (and some exotic) things are covered. The entire presentation is quite elegant. 

I think Kurzweil and Stellmacher's book would serve beautifully as a source for a year-long seminar (or longer?) on group theory at the level of advanced undergraduate students or beginning graduate students.  If I, personally, should wish to learn a lot of serious finite group theory I'd go with this book, perhaps coupled with Rose's well-known Second Course in Group Theory. [Michael Berg]

Paolo Rocchi's The Structural Theory of Probability addresses the question: " What IS probability?" from an angle that, as the author explains, is based on his background in the culture and ideas of software programming. His basis thesis is this: Probability theory from Pascal to Kolmogorov and onwards has focused on events as sets of outcomes or results, and probability as a measure attached to these sets. But this ignores the structure of the processes which lead to the outcomes, and the author explores how taking into account the details of the processes would lead to a more fundamental understanding of the nature of probability.

This is an interesting idea, and the author makes it clear that at present this is a work in process and not yet a finished product, for he says that he has tried to give "an impulse in the right direction" with his theory. Though the concluding chapter gives two examples intended to show how "the structural calculus has a competitive edge over today's calculations," these do not seem to establish the author's points quite as definitively as one would like them to. One hopes that in due course the author will develop his theories further and present overwhelmingly persuasive examples of the advantages of his approach. [Ramachandran Bharath]

I took great pleasure in reading Mathematical Olympiad Treasures, by Titu Andreescu and Bogdan Enescu. This book is the fruit of the prodigious activity of two well-known creators of mathematics problems in various mathematical journals. Andreescu has written many books about olympiad-style problems, several of which have been reviewed here in the past. (See the index of books reviewed by authors!)

Treasures is organized in six chapters. The first three deal with problems from algebra, geometry and trigonometry, number theory and combinatorics. Chapters 4, 5 and 6 give the solutions to problems presented in the first three chapters. In all the chapters, the reader can find numerous challenging problems. All featured solutions are interesting, given in increasing level of difficulty; some of them are real gems that will give great satisfaction to any math lover attempting to solve the problems — or even extend them. I believe strongly that Mathematical Olympiad Treasures will reveal the beauty of mathematics to all students, teachers and all math lovers.

Also by Titu Andreescu (but with Zuming Feng as his collaborator) is A Path to Combinatorics for Undergraduates. This book is an introduction to counting strategies in combinatorial theory. The main mathematical ideas are carefully worked into organized, challenging, and instructive examples given in the nine chapters of this book. In the last chapter we find 111 problems (without solutions). The greater part of them are from various mathematical contests. The big experience of the authors in preparing students for various mathematical competitions allowed them to present a big collection of beautiful problems. By studying this book, undergraduates will be well-equipped to further their knowledge in more abstract combinatorics and its related fields. [Mohammed Aassila]

Though today we know much more about Isaac Newton than we did, say, 50 years ago, he still remains something of a puzzle. Should we think of him as the founder of modern science or as "the last magician," as John Maynard Keynes argued in the 1930s? What are we to make of a man whose writings include extensive material on mathematics, physics, alchemy, and Biblical prophecy?

Isaac Newton's Natural Philosophy opens with a short essay on the history of Newton studies since 1850, highlighting the importance of the careful work on Newton's manuscripts undertaken by several scholars beginning in the 1960s. Of these, perhaps the most impressive is D. T. Whitehead's The Mathematical Papers of Isaac Newton, a monumental — and, at eight huge volumes, massive — work that has still not been absorbed by historians in general. (Alas, this has been allowed to go out of print, which is just too bad.)

This book is a collection of essays by Newton scholars. The first four deal with "motivations and methods," while the remaining six focus on specific aspects of Newton's celestial mechanics and mathematical physics. The last essay is by Richard Westfall (author of the best Newton biography available, Never at Rest). Westfall died before putting the final touches on the essay, so the book also includes a memorial tribute by I. Bernard Cohen.

This affordable paperback is recommended for anyone who is interested in understanding Newton's scientific work in its historical context. [Fernando Q. Gouvêa]

Publication Data

The Theory of Finite Groups: An Introduction, by Hans Kurzweil and Bernd Stellmacher. Springer-Verlag, 2004. Hardcover, xii + 387 pp., $69.95. ISBN 0-387-40510-0.

The Structural Theory of Probability: New Ideas from Computer Science on the Ancient Problem of Probability Interpretation, by Paolo Rocchi. Kluwer Academic, 2003. Hardcover, 176 pp., $140.00. ISBN 030647428X.

Mathematical Olympiad Treasures, by Titu Andreescu and Bogdan Enescu. Birkhauser, 2004. Softcover, vi + 234 pp., $29.95. ISBN: 0-8176-4305-2.

A Path to Combinatorics for Undergraduates: Counting Strategies, by Titu Andreescu and Zuming Feng. Birkhauser, 2004. Softcover, xviii + 228 pages, $39.95. ISBN: 0-8176-4288-9.

Isaac Newton's Natural Philosophy, ed. by Jed Z. Buchwald and I. Bernard Cohen. MIT Press, 2004. Softcover, 376 pp., $22.00. ISBN 0-262-52425-2. Also available in hardcover ISBN 0-262-02477-2.

Michael Berg (mberg@lmu.edu) is professor of mathematics at Loyola Marymount University.

Ramachandran Bharath is visiting professor of mathematics at Colby College.

Mohammed Aassila is a mathematics professor whose research area is analysis. He is interested in mathematics competitions and is the author of two books on the subject: 300 Défis Mathématiques and Olympiades Internationales de Mathématiques.

Fernando Q. Gouvêa is Professor of Mathematics at Colby College, editor of FOCUS and FOCUS Online, and co-author of Math through the Ages.

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Read This! is the MAA Online book review column. Contributions are welcome; contact the editor if you'd like to be one of our reviewers. Books for review should be sent to the editor: Fernando Gouv&ecric;a, Dept. of Mathematics, Colby College, Waterville, ME 04901. Publishers, please check our reviews information page.

MAA Online is edited by Fernando Q. Gouvêa (fqgouvea@colby.edu).
Last modified: Thu Jul 29 10:51:11 Eastern Daylight Time 2004