Maximal subgroups of the Monster

From time to time people ask me whether the maximal subgroups of the Monster are known. The reason for this is that in 1980 I began work on the maximal subgroup problem for the sporadic simple groups, one after another, and dreamed of the prospect of making this my life’s work, completing the final case (the Monster) just in time for my retirement in 2023. But I worked too fast, and after 20 years, only the Monster was left. By that point, my students and I and other collaborators (mostly the students) had largely reduced it to calculations that we estimated were feasible in practice.

Contrary to what most people think, it is not a mathematician’s job to do calculations. A mathematician’s job is to show how to avoid calculations. Of course, in order to do this it is usually necessary to understand thoroughly the calculations that one wishes to avoid. In this particular case, it was clear from decades of experience that most of the calculations could not be avoided. That is not to say that one or two of them could not be avoided by suitable trickery, and we did write some papers explaining some of this trickery. But it is ultimately just calculation, which (again, contrary to most people’s expectations) most mathematicians are not very good at.

The calculational infrastructure that we were using was built 20 years ago by amateurs. It is well past its use-by date, and the only reason you could expect me to complete the job is if you thought I had nothing better to do. You may well think that, but I don’t. And if you want to know what the maximal subgroups of the Monster actually are, reliably, for the benefit of future generations, the last thing you should do is entrust the job to a retired professor working with inadequate tools, who doesn’t care what the answer is.

First you need some proper infrastructure. Get a professional computer scientist to do it – for example, Martin Seysen, who has in fact already done this, several orders of magnitude faster than we did. Then get an enthusiastic student to do the calculations, (a) so that they can learn the nitty-gritty of research, and (b) so they can get the kudos of having finished off the maximal subgroup problem for sporadic groups. Above all, find someone who cares. I don’t.

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9 Responses to “Maximal subgroups of the Monster”

  1. Robert A. Wilson Says:

    This is why mathematics is called the Queen of the Sciences: she doesn’t do anything herself, but she tells everyone else what to do.

    • Robert A. Wilson Says:

      And they have to obey, because mathematics is an absolute monarch, not a constitutional monarch.

    • Robert A. Wilson Says:

      The bolsheviks (physicists) think they can depose mathematics from the throne, and rule in her stead. They are wrong. If you try to impose laws on the universe that contradict the laws of mathematics, as indeed physicists do, then the universe will refuse to obey those laws, and, ultimately, there is nothing the bolsheviks can do about it.

  2. Robert A. Wilson Says:

    A correspondent objected to me saying that a mathematician’s job is not to do calculations, thereby illustrating my point perfectly. The job of calculating is and always has been given to a computer. My father was a mathematician, and my mother was a computer, so I know the difference.

  3. mitchellporter Says:

    Monster group theory is how I first heard of you, many years ago.

    You’re probably aware that Witten wrote a paper investigating whether the “moonshine module” field theory of Frenkel, Lepowsky, and Meurman, might be holographically equivalent to pure gravity in three spacetime dimensions… Did this work ever influence your own reflections? As far as I know, even just within string theory, it’s still something of an orphan result. There’s a technical reason why the Monster would lead to a string theory vacuum with no gauge fields, but there’s no deeper conceptual explanation for why the largest sporadic would have such a direct relationship to one of the basic forces.

    • Robert A. Wilson Says:

      The simple answer is “no”, but it’s really much more complicated than that. I was around in the early days of Monstrous Moonshine, and saw a stream of physicists come to talk to Conway about it. He himself was dubious about any physical connection, but Borcherds made more of it. I remember a conference/workshop on Moonshine in Edinburgh in 2004, attended by many of the big names, where I tried to get to grips with why physicists were interested in Moonshine, but largely failed. There was another workshop in Durham in 2015 on moonshine and string theory, which I attended, which made even less sense.

      Meanwhile, I had got interested in E8, and how Lisi and others were trying to link it to physics, and attended the Banff workshop in 2010, where we explored possible connections between 3×3 octonion matrices for F4/E6 and for the Conway group. So there was a time at which I considered the possibility that the Conway group and the Leech lattice might describe a discrete 3-dimensional universe, with the quantum fields encoded in the octonions.

      But by 2015 I was pretty much convinced that none of these ideas could ever work. That was when I went back to first principles, and started to attack the weak points in the foundations. It has taken me a while to complete the full structural survey, but it is now clear the whole edifice in a dangerous state, as it has no foundations at all. Any small earthquake or winter storm could bring it crashing to the ground in an instant.

      • Robert A. Wilson Says:

        Not to mention the possibility that the whole standard model might just get washed away by the Ricci flow, or a flood of new ideas coming down from the melting glaciers on the mountain tops. Or it could simply get blown up by an exploding undersea volcano in Tonga that not only destroys the new land that connected quantum mechanics and gravity, but blows up half the original islands themselves.

      • Robert A. Wilson Says:

        I think there is not much doubt now that the climate (in theoretical physics) is changing, and that the temperature has been rising steadily for a century or more. We cannot be sure what exactly will happen, but more extreme theories are likely to arise. All those beautiful snowflake theories that have been frozen out in the ice will start to melt and make themselves known. The left-handed and right-handed spins of cyclones and anticyclones will continue to gain energy and blow away not just the cobwebs, but all the dead wood as well. All those beautiful tapestries woven out of string will be swept away, along with the buildings that housed them, and the Marie Antoinettes that admired them, their heads still full of useless ideas like “let them eat cake”.

        You cannot build a theory of the universe out of an idea like “let them eat cake”, which appears to be the sum total of string theory at this point.

      • Robert A. Wilson Says:

        The trouble with physics is it is all cake and no bread. So when I come along, offering freshly baked bread instead of very stale cake, it is spurned.

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