Symmetry and the Monster: One of the Greatest Quests of Mathematics [ペーパーバック]

According to the blurb on the back, the American Mathematical Monthly described this book as "truly a page-turner". I have to say it is not.

Mark Ronan's task is to take us through the history of group theory culminating in the recently-completed project to classify the finite simple groups. This has taken decades of work by large numbers of highly-skilled mathematicians, with proofs so long and abstruse that there is a genuine concern that no future generation of mathematicians will be able to comprehend them.

How do you communicate this to a lay audience? The key decision for the writer is to gauge his audience. Ronan's view is a readership which knows no group theory. He therefore can't even define a simple group: "a simple group is a group which is not the trivial group and whose only normal subgroups are the trivial group and the group itself" - Wikipedia.

The reader, lacking help in engaging with the subject matter, is instead entertained by concise and amusing mini-biographies and anecdotes about the many participants in the quest. Ronan is a little dry as a writer, but in general this works well enough, although he is too indulgent of such monstrous personages as Sophus Lie. The final milestone in the classification project was confirmation of discovery of the mathematical Monster, the largest of the 26 sporadic groups. This was big news even on conventional news outlets, such as the BBC.

In conclusion, this book will work for mathematicians who know some group theory and who like the historical context spelled out. I don't think many people not educated in mathematics will make it through to the end. With this in mind, Ronan could have profitably added a chapter at the beginning (or even an appendix) where he took the reader through normal subgroups, quotient groups and on to simple groups. He would then have been able to use correct terminology (his own merely irritates) and the journey would have been a lot more satisfying. Perhaps for the second edition?

While it is simple enough to conceive an object in one, two or three dimensions, adding just one more dimension can be mind-bending. The four dimensional cube - or tesseract - cannot be truly perceived, but we can at least get a glimmer of it when we look at its projection, which appears like a cube within a cube. Five dimensions are even harder to perceive. The Monster, the subject of Mark Ronan's Symmetry and the Monster, has 196,884 dimensions. It seems appropriately named.

What is the Monster, however? This takes a while to describe, and it all begins with the brilliant Galois, a mathematical genius who would be dead by 20 after being on the losing side in a duel. Galois would make some major strides in the field of algebra known as group theory. A group is really just a self-contained set of numbers (or other components) with an operation (such as addition) and certain properties (such as closure, the idea that when you do the operation on two members of the set, you get another member of the set; for example, with the whole numbers and addition, adding any two positive integers gets you another positive integer).

Groups can be both finite and infinite, and among finite groups, there are so-called simple groups (or what Ronan calls atoms of symmetry). These are not simple as in easy, but simple as they cannot be deconstructed into simpler groups, just as when you factor a number, you cannot factor any further when you reach the prime factors. Most simple groups fit into certain families, but there also 26 exceptional groups (or sporadic groups). Determining that the number was 26 and finding all these groups is what Symmetry and the Monster is all about. The final group would be the biggest, by far: the Monster.

Perhaps the best book dealing with the solution of a tough problem is Simon Singh's Fermat's Enigma, dealing with the proof of Fermat's Last Theorem. Ronan's book is not as easy of a read, but then again, he has a tougher row to hoe: while Fermat's Last Theorem is relatively easy to understand (though difficult to prove), the concept of symmetry groups is a bit more esoteric. Operating within this constraint, Ronan does a good job, writing clearly, with both a sense of history and sense of humor. This is not an easy subject to really grasp, but it may be ultimately rewarding to those who stick with it.

This book is a ripping yarn - I couldn't put it down. My wife asked when I came to bed at midnight : "Maths porn again darling? " Although I have done some group theory, my knowledge was nowhere near enough to make any meaningful attempt to understand the detail of the Monster project and like many others, it remained an intractable beast that others were battling with. Ronan explains in a high level way the history to the Monster starting with Galois and working his way through the historical development. He peppers the account with all manner of interesting observations about the participants which are revealing in ways that one does not often find in maths books. For instance, there is a revealing comment on page 152 about someone of the stature of John H Conway who confessed he "felt like a fraud" in giving talks early on in his work on Monster. It seems a graduate student asked him the obvious question namely " How do you now that your new group can't be decomposed into something simpler?" Maths is an unforgiving business.

Mark Ronan who has worked with and/or knows most of the heavy hitters in the field has done a wonderful job explaining the history of what is an extraordinary undertaking not only in purely intellectual terms but also in personal terms. The sociological dimensions of this immense task are reflected in all manner of small and large stories. Thus John H Conway bargains with his wife to have blocks of time away from the 4 kids so he can crack some problems and he manages in 12  hours to prove something important about the Leech Lattice. That set him up for life. The proofs in this field can be hundreds of pages long - one by Mason is 800 pages long and has not been published. This itself imposes huge strains on referees. The classification task (which I had read about but had no detailed knowledge of what was involved other than a vague idea it was the equivalent of the 30 Years War) demonstrates what a small group of intensely committed people can do. What they were doing was to provide a set of knowledge that subsequent mathematicians could understand given that the barriers to entry to the detailed knowledge are so high.

At a purely personal level one has to marvel at how some of the people concerned threw their lot in with this "monstrous" task. Every budding PhD students knows that problem selection is important and it does not pay to spin one's wheels forever on some obscure problem.

There are some truly astonishing connections revealed in this book. The connection between the number theoretic j function and the character set of the Monster (see pages 192-193) is remarkable but then there is the even more remarkable connection between light rays and the Leech Lattice (see page 224).

Mark Ronan has done a great service to all those who have served and still served in the battle with the Monster. Most of the main workers in the field are no longer with us so Ronan's book provides the general community with some sense of their achievements.

For those interested in Lie Theory may I suggest John Stillwell's accessible book "Nave Lie Theory" as a starting point. He strips a way a lot of the overheard that makes Lie Theory so daunting.

Peter Haggstrom
Bondi Beach
Sydney, Australia