Symmetry and the Monster by Ronan Mark

Author:Ronan, Mark
Language: eng
Format: epub
Publisher: Oxford University Press, UK
Published: 2006-08-09T16:00:00+00:00

Here, the authors proved a famous conjecture, to the effect that all finite [symmetry atoms] have even size. I am not sure who was the first to observe this. Fifty years ago it was already referred to as a very old conjecture ... [but] nobody ever did anything about it, simply because nobody had any idea how to get started.*

After proving a theorem like this, what do you do for an encore? The whole point of the Feit–Thompson theorem was to open the way to classifying all symmetry atoms, so one obvious thing was to set about this task. In the early days, Thompson was overheard saying he would knock this off in short order, but there was no such luck. It turned out to be very complicated indeed because of the exceptions, which eventually led to the Monster, but more on that later.

Remember what the idea was. You take a cross-section in something you know about, and show that there is nothing else having this cross-section. Brauer had already dealt with some families, but there was still plenty of work to do. After the special year at the University of Chicago, Thompson went to Harvard to finish writing the proof of the Feit–Thompson theorem, and work in the presence of Brauer.

In 1962 Thompson came back to a position in Chicago, and started work on cross-sections of type A1. One class of these arose in a special family of symmetry atoms, but the rest were not cross-sections in any known symmetry atom. Thompson wanted to prove that this was the end of the story, so he took an imaginary symmetry atom having a cross-section of type A1, and tried to show that either it was in the family he wanted, or it led to a contradiction.

Now you might suppose that cross-sections of type A1, rather than those of higher rank like A2 or A3, ought to be relatively easy. But this is not the case at all. Think of a monocycle, a bicycle, and a tricycle as analogues to type A1, A2, and A3. The monocycle is the trickiest to deal with – and you can do things with it that are impossible with bicycles and tricycles. It is that way in mathematics – the low rank cases are trickiest and that is precisely where unusual things can happen. Thompson worked very hard on the problem, and eventually wrote up his results.

He hadn’t yet prepared them for publication, and was working on other things, when in 1964 he received a letter from a mathematician named Zvonimir Janko in Australia. Janko had approached the same problem in connection with his own work, and had found that when the cross-section was the smallest symmetry atom in the A1 family he couldn’t get the required contradiction. Alperin remembers the occasion: ‘I remember it very clearly. Thompson told me of the letter at tea, and he was smiling about it. The next morning he wasn’t smiling.’

Thompson had already replied to Janko, but immediately after posting the letter he noticed an error in his own argument.

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