The Classification of the finite simple groups, leading to the Monster and Moonshine

Each finite group can be deconstructed into a collection of finite simple groups, which I call finite 'symmetry atoms' in my book Symmetry and the Monster. All finite simple groups have been found, and each one either fits into a well-understood table, in which case it is called a 'group of Lie type', or it is one of 26 exceptions, called sporadic groups.

The long process of finding all finite simple groups, and showing there are no more, got seriously underway in the early 1960s with the proof of the Feit-Thompson theorem, due to the American mathematicians Walter Feit and John Thompson. They proved that if a finite simple group was not generated by a single element, then it must contain an operation of order 2. Such operations yield subgroups that I have called cross-sections (the technical term is involution centralizer), and from work of Richard Brauer it was theoretically possible to reconstruct a finite simple group from its cross-sections. This suggested a way of classifying them all by analysing their possible cross-sections. Most groups could not be cross-sections in any finite simple group, but for those that could it was necessary to find all simple groups having such cross-sections. Usually there was no more than one, but several new simple groups were discovered in this way, and if a new one came on the scene, mathematicians had to check whether it could appear as a cross-section in anything not previously known.

This project was extremely complex, but in 1972 the American mathematician, Daniel Gorenstein proposed a 16-step plan for its completion. At that time it was remarkable that anyone could see a way through to the end, but Gorenstein was a great optimist and helped organise and orchestrate the teamwork necessary to cover all cases. By the mid-1970s things were moving rather fast, particularly since an American mathematician named Michael Aschbacher started short-cutting a lot of the planned steps in Gorenstein's program. By about 1980 most experts felt we had a complete list of finite simple groups. But there was always the question of errors, and writing in a popular article at about that time, the English mathematician, John Conway reports that someone asked him about this, and whether he was an optimist or a pessimist.

I replied that I was a pessimist, but still hopeful, and was delighted to find that this answer was misinterpreted in exactly the way I had maliciously desired!

Among those who are engaged in the great cooperative attempt to classify all the finite simple groups, 'optimism' usually describes the belief that there are no more such groups to be found, since new groups appear as obstacles in the path of progress. My own view is that simple groups are beautiful things, and I'd like to see more of them, but am reluctantly coming around to the view that there are likely to be no more to be seen.

A conviction that there really was nothing new to be found grew stronger as some of the previous results were analyzed in a revision of the classification proof. However, one massive 800-page manuscript had never been published, and created an awkward gap. This was the 'quasi-thin' case, and although there seemed to be nothing new here either, the arguments contained gaps, and subsequent work that might have covered the same territory had fallen short. Papers like the quasi-thin manuscript were usually written in the spirit of closing things off, and if a particular situation led to a contradiction, then that settled the matter. However, some contradictions were chimeras. They didn't really exist, and as Conway wrote in 1980:

Quite a large number of the groups . . . [were] constructed after somebody had already proved them impossible! When David Wales and I set out to construct the Rudvalis group, for example, we soon ran into a contradiction which refused to go away even after we had condensed it onto one side of a sheet of paper and scrutinised it for several days. Fortunately we were so convinced that the group existed that eventually we just put that piece of paper aside and constructed the group by another method that carefully went nowhere near our contradiction! Another group theorist later told me that he too had disproved the Rudvalis group, although he had only used the assumption that it contains a subgroup that it does, in fact, contain! . . . What worries me is the nagging thought that another group like the Rudvalis group might have been disproved somewhere in the classification programme by someone who had no overwhelming conviction that it existed.

The trouble is that groups behave in astonishingly subtle ways that make them psychologically rather difficult to grasp. We might say that they are adept at doing large numbers of things well before breakfast.

The quasi-thin case remained an awkward gap until Michael Aschbacher and Stephen Smith decided to tackle it head on. It was a massive project, and their work now occupies 1200 pages in two-volumes published in November 2004. It shows that there is nothing new in quasi-thin territory.

The other major project — the Revision — was started in the 1980s by Gorenstein, along with Richard Lyons and Ronald Solomon who have been carrying it forward since Gorenstein's death in 1992. Their aim is to present all the arguments in a coherent way, a project they hope to complete by 2010. This is important work because the Classification was so technical that it required years of study in order to follow the arguments, and these arguments themselves were sometimes written in haste, and were not user friendly, to put it mildly. They were concise, sometimes to the point of being inaccessible, even to experts in the field. If the whole thing were not rewritten it was feared that future generations of mathematicians might find it incomprehensible.