The Mathieu group M24
John Baez
and right now I'm trying to learn more about the Mathieu group M24,
which Conway called the "most remarkable of all finite groups."
This group is the automorphism group of the "Steiner triple system"
S(5,8,24) - that is, a setup with 24 points with certain distinguished
8-element sets called "octads", such that each point is in 5 octads.
I'm wondering how the number 24 here is related to its appearance
in other contexts, like Monstrous Moonshine. In particular, is there
a nice "geometrical" intepretation of S(5,8,24) where I can see the
24 points as something nice like the vertices of the 24-cell, or
24 directions in the Leech lattice, or something like that?
I just found this reference which I guess I should read:
R. T. Curtis, Geometric interpretations of the 'natural' generators
of the Mathieu groups, Math. Proc. Camb. Phil. Soc. 107 (1990), 19-26.
Does anyone here know what it says? This website:
http://www.math.uic.edu/~fields/DecodingGolayHTML/conclusions.html
says:
Curtis describes natural "geometric" generators for the Mathieu groups M12
and M24. M12 (the automorphism group of the extended ternary Golay code of
length 12) is constructed from the group of a dodecahedron with certain
additional "twists".
The Mathieu group M24 is also constucted geometrically, but in terms of a
rather complicated figure (an icosatetrahedron) having 24 heptagonal faces,
56 vertices and 84 edges. If we consider just the vertices and edges, we
have a non-planar graph that can be drawn (without crossing lines) on a
surface of genus 3. Since this group and the extended binary Golay code
are intimately related, it seems natural (in view of our construction)
to consider whether there are natural geometric generators for M24 that
can be viewed via the dodecahedron.
I know about how M12 is related to the dodecahedron:
http://math.ucr.edu/home/baez/week20.html
but what about M24?
Nick Patterson
I wrote my thesis (too many years ago!) on the symmetry
group of the Leech lattice (big Conway group .0)
If you know how the Leech lattice is defined there is an
"obvious" subgroup of symmetries 2^12.M24
(Change sign on a "codeword" of the Golay code and apply
a permutation of M24 to the coordinates)
So M24 is acting to permute the coordinates of the Lattice.
To construct the Big Conway group you have to find in addition
another symmetry. This is what Conway did. After that it is
fairly easy to find the order of the group and to prove that
(after factoring out the central inversion) the group is simple.
24 and moonshine will be harder.
Incidentally I just heard a talk by Conway where he mentioned
interest of physicists in moonshine. Physics students are not
traditionally so great at modular forms, and Conway doesn't
claim to be a physicist. He described his conversations with
physicists as "the dumb leading the blind"
Your first reading should be Conway: Three lectures on finite simple groups
where he masterfully builds .0 from remarkable permutation representations
of PSL(2,n) on n points for n small.
Nick Patterson
>In article <_ITs9.5212$US2....@vixen.cso.uiuc.edu> you write:
>I'm finally starting to study the sporadic finite simple groups,
>and right now I'm trying to learn more about the Mathieu group M24,
>which Conway called the "most remarkable of all finite groups."
>
>This group is the automorphism group of the "Steiner triple system"
>S(5,8,24) - that is, a setup with 24 points with certain distinguished
>8-element sets called "octads", such that each point is in 5 octads.
>
>I'm wondering how the number 24 here is related to its appearance
>in other contexts, like Monstrous Moonshine. In particular, is there
>a nice "geometrical" intepretation of S(5,8,24) where I can see the
>24 points as something nice like the vertices of the 24-cell, or
>24 directions in the Leech lattice, or something like that?
>
>I just found this reference which I guess I should read:
>
>R. T. Curtis, Geometric interpretations of the 'natural' generators
>of the Mathieu groups, Math. Proc. Camb. Phil. Soc. 107 (1990), 19-26.
--F219B48A2F.1035211208/mail1.panix.com--
Greg Kuperberg
John Baez <ba...@math.removethis.ucr.andthis.edu> wrote:
>I'm wondering how the number 24 here is related to its appearance
>in other contexts, like Monstrous Moonshine. In particular, is there
>a nice "geometrical" intepretation of S(5,8,24) where I can see the
>24 points as something nice like the vertices of the 24-cell, or
>24 directions in the Leech lattice, or something like that?
Conway has a very appealing set of lectures on this in Conway and Sloane,
"Sphere Packings, Lattices, and Groups".
--
/\ Greg Kuperberg (UC Davis)
/ \
\ / Visit the Math ArXiv Front at http://front.math.ucdavis.edu/
\/ * All the math that's fit to e-print *
Noam D. Elkies
and Greg Kuperberg, I add:
1) The observation that Conway's lectures on sporadic groups
are in fact contained as a series of three chapters in
_Sphere Packings, Lattices, and Groups_;
2) The intriguing connection between M24 and automorphisms of K3 surfaces,
noted in S. Mukai: "Finite groups of automorphisms of K3 surfaces
and the Mathieu group", Invent. Math. 94 (1998) 183--221 [MR 90b:32053].
--Noam D. Elkies
John Baez
acts to permute some set of "coordinate axes" in the Leech lattice.
This is exactly the sort of thing I wanted.
In article <_ITs9.5212$US2....@vixen.cso.uiuc.edu>,
John Baez <ba...@math.removethis.ucr.andthis.edu> wrote:
>This group is the automorphism group of the "Steiner triple system"
>S(5,8,24) - that is, a setup with 24 points with certain distinguished
>8-element sets called "octads", such that each point is in 5 octads.
I slipped here - I should have said:
This group is the automorphism group of the "Steiner triple system"
S(5,8,24) - that is, a setup with 24 points with certain distinguished
8-element sets called "octads", such that any 5-element set of points
is contained in a unique octad.
Greg Kuperberg
>This group is the automorphism group of the "Steiner triple system"
>S(5,8,24) - that is, a setup with 24 points with certain distinguished
>is contained in a unique octad.
Actually it's a Steiner system, but not a Steiner triple system.
The latter is specifically of the form S(2,3,n).
Brendan McKay
It is a "Steiner system" but not a "Steiner triple system". The word
"triple" refers to Steiner systems consisting of 3-element subsets,
which form the most studied and most understood class of Steiner
systems. I'm sure this terminology is near-universal amongst
design theorists.
Brendan.