Mark Ronan's website
The Classification of the finite
simple groups
Each finite group can be
deconstructed into a collection of finite simple groups, which I refer to as 'finite symmetry atoms' in my book Symmetry and the Monster. All finite simple groups have been found, and each
one is either a group of Lie type, and therefore fits into a well-understood
table, 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 rotation, 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 ways that could not be described as user friendly.
They were concise, sometimes to the point of being inaccessible to anyone but
an accomplished expert. Without a rewriting of the whole thing it was feared
that it might be incomprehensible to future generations of mathematicians.