28
votes
Why does the monster group exist?
[comment turned into an answer, as suggested by Mikhail Katz]
a 2022 paper by Scott Carnahan, "Why do the symmetries of the monster vertex algebra form a finite simple group?" goes some way ...
- 188k
26
votes
$H^4$ of the Monster
In arXiv:1707.08388, I calculate that the cohomology class you described has order 24 and that it is not a characteristic class in the ordinary sense.
- 54.6k
23
votes
What is the geometric shape of the Monster sporadic group?
In the penultimate chapter of Sphere Packings, Lattices and Groups, the authors define a $196884$-dimensional real vector space and a faithful representation of the Monster group on that space.
Now, ...
- 12.4k
16
votes
Accepted
71, the Monster, and c = 24 CFTs
Schellekens' enumeration is exhaustive in the following sense: the degree 1 subspace of the meromorphic CFT/vertex algebra is naturally a Lie algebra, and it is known that this Lie algebra must be one ...
- 176
16
votes
Accepted
What is the geometric shape of the Monster sporadic group?
It is possible that Conway was referring to the generic construction that works for all finite groups equipped with faithful representations, given in the other answers. However, I think it is more ...
- 45.7k
16
votes
Accepted
Why does the monster group exist?
The OP wrote:
I am trying to understand what he means by "why".
Although several respondents have expressed skepticism that this question can be answered satisfactorily, I maintain that it'...
- 82.7k
15
votes
Accepted
Is $J_1$ a subquotient of the monster group?
No $J_1$ is not involved in (i.e. is not a subquotient of) the Monster.
The six sporadic simple groups listed here as "Pariahs" are precisely those that are not involved in the Monster, namely $J_1$, ...
- 37.4k
10
votes
What is the geometric shape of the Monster sporadic group?
It is not too surprising that the Monster group $M$ is the symmetry group of something geometrical.
E.g. every group is the group of symmetries of some convex polytope.
You can even make it a vertex-...
- 13.6k
9
votes
Why do these two Monster-related calculations yield $163$?
If there is a connection, I would expect it to be related to the fact that the $j$ invariant is connected to both Moonshine and to class field theory. Note that 7, 11, 19, 43, 67, and 163 are all the ...
- 11.1k
6
votes
Where can I find a table of the exponents of the sporadic groups?
I couldn't find an online table of exponents for sporadic groups, so I used GAP to produce one:
$$
\begin{align*}
\mathbf{Group}&&\mathbf{Exponent}&&\mathbf{Factorization}\\
M_{11}&...
- 161
4
votes
Accepted
Where can I find a table of the exponents of the sporadic groups?
From the comments,
This information can be calculated easily from the printed character tables in the ATLAS of Finite Groups (which include orders of elements in conjugacy classes) or, perhaps more ...
- 373
3
votes
Accepted
On level $10$ of the McKay-Thompson series of the Monster
For $s_{10C}$, Maple finds this $7$-term recurrence:
...
- 41.1k
2
votes
Computing Thompson series for the monster group
In the OEIS index, see
"McKay-Thompson sequences or series for Monster simple group, sequences related to"
These may come with tables of coefficients, and with programs to compute ...
- 41.1k
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