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The Monster group is the largest of the sporadic simple groups, and has been proven by Wilson to also be a Hurwitz group. It has a presentation in terms of Coxeter groups, specifically Y443 along with the "spider" relator, and quotienting out by the center. However, I am interested in a presentation as a Hurwitz group.

Specifically, what is a presentation of the monster in terms of two elements a and b such that a has order 2, b has order 3, and ab has order 7? Also, what is the smallest possible order of the commutator of a and b in such a presentation?

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  • $\begingroup$ It doesn't look like this is known, at least none of the papers citing Wilson's paper seem to do it. I'm not an expert though so I suppose I could have missed something. $\endgroup$
    – Noah Snyder
    Commented Jun 23, 2020 at 0:30
  • $\begingroup$ I would guess that it would be possible to compute such a presentation by applying a standard change of generator algorithm to the existing presentation, although the resulting presentation would be unlikely to be particularly illuminating. To do that you would need to be able to do basic computations with elements of the Minster, but that is possible - Wilson has software for that. Wilson would also be the best person to ask about the minimal order of $[a,b]$. $\endgroup$
    – Derek Holt
    Commented Jun 23, 2020 at 7:59
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    $\begingroup$ Actually, the last question does not depend on a presentation, but just on a generating subset $(a,b)$ satisfying the $(2,3,7)$ relations (presentation refers to an explicit set of relations). Whether there's a "short" presentation is intriguing, anyway. $\endgroup$
    – YCor
    Commented Jun 23, 2020 at 9:26
  • $\begingroup$ I suppose such a presentation gives a closed surface with the Monster group acting on it, but it will be pretty big. $\endgroup$
    – Ian Agol
    Commented Sep 14, 2022 at 13:18
  • $\begingroup$ A related question: is there a presentation by three involutions whose products have order 2, 3, and 7 respectively? $\endgroup$
    – Ian Agol
    Commented Sep 14, 2022 at 17:25

1 Answer 1

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I have computed two pairs of generators $(a,b)$ of the Monster satisfying the relations $a^2 = b^3 = (ab)^7 = 1$ using [1]. In both cases $a$ is of class 2B, $b$ is of class 3B in the Monster, and the commutator $[a, b]$ has order 39. This gives an upper bound for the minimal order of that commutator. More details of the computation are given in subdirectory applications/Hurwitz of [1].

[1] Martin Seysen, the python mmgroup package,

https://github.com/Martin-Seysen/mmgroup

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  • $\begingroup$ This is amazing, thank you! I've had a look through the comments on the github package, that does seem to take quite a while to find a Hurwitz generating set. Are calculations for that sill ongoing? I'd be keen to see if there are other possible generator pairs, with a different order of the commutator. $\endgroup$
    – Thomas
    Commented Sep 17, 2022 at 5:59
  • $\begingroup$ Computations are still ongoing. I expect to find about one pair per week, and a friend of mine may find about one pair per day with a faster computer. I have not yet looked for pairs $(a,b)$ of classes (2B, 3C), which may exist according to the paper The Monster is a Hurwitz group by R.A. Wilson, $\endgroup$
    – Martin Seysen
    Commented Sep 17, 2022 at 11:45
  • $\begingroup$ Have any new pairs been found? @Martin Seysen $\endgroup$
    – Thomas
    Commented Sep 30, 2022 at 7:12
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    $\begingroup$ Altogether I found 5 pairs pairs $(a,b)$ of classes (2B, 3B). The commutator has order 39 in all cases. $\endgroup$
    – Martin Seysen
    Commented Sep 30, 2022 at 10:00

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