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I was reading about the monster group, and how hard it was to do calculations in it, and I wondered: Is there a known presentation of the monster group? I know that it is a hurwitz group, but other than that I don't know. If we have two generators a and b such that $a^2=b^3=(ab)^7=1$, what are the possible orders of $[a,b]$? I believe that the conjugacy classes are known, so if you are given the conjugacy classes of two elements x and y, can we determine the possible conjugacy classes of $xy$?

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4 Answers

up vote 16 down vote accepted

There's a 12-generator 80-relator presentation for the Monster group. Specifically, we have 78 relators for the Coxeter group Y443:

  • $12$ relators of the form $x^2 = 1$, one for each node in the Coxeter-Dynkin diagram;
  • $11$ relators of the form $(xy)^3 = 1$, one for each pair of adjacent nodes;
  • $55$ relators of the form $(xy)^2 = 1$ (commutators), one for each pair of non-adjacent nodes;

together with a single 'spider' relator, $(a b_1 c_1 a b_2 c_2 a b_3 c_3)^{10} = 1$, which results in the group $M \times C_2$. We can get rid of the $C_2$ by quotienting out by an eightieth relation, $x = 1$, where $x$ is the unique non-identity element in the centre of the group.

See http://www.maths.qmul.ac.uk/~jnb/web/Pres/Mnst.html for the explicit Coxeter-Dynkin diagram.

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So, it is x^2=1 for each node, (xy)^3 for each pair of adjacent nodes, and [x,y]^2 ([x,y] is the commutator of x and y) for each pair of non-adjacent nodes. Then the "spider" relation, and quotienting out the center. Wow, that is very interesting. –  Thomas Sep 16 '13 at 2:00
    
Also, what is the center element? It doesn't say on the page. –  Thomas Sep 16 '13 at 2:08
    
What would happen if you did not include the spider relation? –  Thomas Sep 16 '13 at 3:58
    
without the spider, you get some weird infinite Coxeter group. –  Dima Pasechnik Sep 16 '13 at 6:54
    
Your first comment is correct, except that the relators corresponding to non-adjacent nodes are $[x,y] = 1$ (i.e. $x$ and $y$ commute), which is equivalent to $(xy)^2 = 1$ since $x$ and $y$ are elements of order 2 (Coxeter groups are generated by reflections). –  Adam P. Goucher Sep 16 '13 at 8:22
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A presentation for $M$ was constructed by A.A.Ivanov. This presentation is not so easy to describe; it arises from an amalgam of parabolics in certain diagram geometry. S.Norton then proved that the Y-presentation from ATLAS indeed describes the "Bimonster" (see Derek Holt's comment to the previous answer).

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I never would have thought it would be the wreath product of the monster by Z2. The link you gave me does not let me see where the monster group is derived, let alone the whole chapter. Could someone email me a copy? –  Thomas Sep 15 '13 at 14:03
    
sure, but where to? your profile doesn't reveal any email... –  Dima Pasechnik Sep 15 '13 at 17:21
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Thank you for the paper. –  Thomas Sep 16 '13 at 2:01
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Googling "monster group as a hurwitz group", the first hit is web.mat.bham.ac.uk/R.A.Wilson/pubs/MHurwitz.ps

which I believe contains the answer.

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I don't think that it actually gives a presentation for the group, it just proves that it is a hurwitz group, and it shows that the conjugacy classes of a, b, and ab are 2B, 3B, 7B, respectively. –  Thomas Sep 15 '13 at 10:21
    
Actually, I wouldn't mind if the presentation was not in hurwitz group form. I just want to know if there is a presentation. Is the monster group the group generated by a and b and satisfying those relations (in the Atlas page mentioned before your edit), or is it just a quotient? –  Thomas Sep 15 '13 at 10:24
    
The presentation referred to in the ATLAS, as a Coxeter group with extra relations, is actually a presentation of the "Bimonster" $M \wr C_2$ rather than of $M$ itself. It was eventually proved correct by Ivanov (I remember that being announced). I am not sure whether anyone has found an explicit presentation of $M$. –  Derek Holt Sep 15 '13 at 10:55
    
@Derek: surely, one of main achievements of the theory of diagram geometries is construction of explicit presentations for most sporadic simple groups (perhaps their automorphism groups, sometimes). E.g. such a presentation is known for $M$. –  Dima Pasechnik Sep 15 '13 at 11:57
    
@Dima Pasechnik: Yes, you are right, it's in the paper of Ivanov, which you can find by googling:diagram geometry presentation monster –  Derek Holt Sep 15 '13 at 14:13
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Bimonster can be presented by the Coxeter group of the 26-node graph of the projective plane of order 3: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.23.2634 (26 Implies The Bimonster, by John H. Conway and Christopher S. Simons). See also http://citeseer.uark.edu:8080/citeseerx/viewdoc/summary;jsessionid=57D5F004B2D42A5AE3CF28FD1C29E8E7?doi=10.1.1.137.1363 (An Elementary Approach to the Monster, by Christopher S. Simons).

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