# Questions tagged [monster]

Questions about the Monster group, the largest of the sporadic simple groups. This group acts as symmetries on a vertex operator algebra whose graded dimension is the elliptic $j$-function.

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### Why does the monster group exist?

Recently, I was watching an interview that John Conway did with Numberphile. By the end of the video, Brady Haran asked John:
If you were to come back a hundred years after your death, what problem ...

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### On level-$12$ of the McKay-Thompson series of the Monster and the Domb numbers

(This continues from level 10.) Given some moonshine functions $j_{N}$. There are nice descending and consistent relations for levels $6m$ with $m= 2,3,5,$
$$j_{12A} = \left(\sqrt{j_{12H}} + \frac{\...

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### On level $10$ of the McKay-Thompson series of the Monster

(For brevity, the level-6 functions have been migrated to another post.)
I. Level-10 functions
Given the Dedekind eta function $\eta(\tau)$. To recall, for level-6,
$$j_{6A} = \left(\sqrt{j_{6B}} + \...

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### Is there an extension of Ogg's results to surfaces of Genus 1

The first hints of moonshine appeared around 1974 when Andrew Ogg noticed that quotienting the hyperbolic plane by normalizers of the Hecke Congruence subgroups $\Gamma_{0}(p)$ has genus zero iff p is ...

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### Kissing the Monster, or $196,560$ vs. $196,883$

The $D = 24$ kissing number is $196,560$, and the dimension of the smallest non-trivial complex representation of the Monster group is $196,883$. These two numbers are nearly but not quite equal, and ...

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### Monster group as automorphism group of a distributive lattice

It is known that every finite group is the automorphism group of a finite distributive lattice.
Question: What is the minimal order of a distributive lattice $L$ such that the automorphism group of $...

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### Is $J_1$ a subquotient of the monster group?

Edit: I was able to make a 3D diagram of the happy family if anyone is interested!
https://www.youtube.com/watch?v=_4IjnIcECoQ
I'm working on a twitter thread about the monster group, because I saw ...

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### What is the geometric shape of the Monster sporadic group?

Conway made the comment that the Monster group represents the symmetries of a shape in 196,883 dimensions, something like a "star you hang on a Christmas tree."
My question is, What do we know (or ...

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### Where can I find a table of the exponents of the sporadic groups?

Is there a table showing Sporadic Groups and their exponents, and, perhaps, other basic properties.
In particular, I'm interested in what the exponent of the Monster Group is. (Obviously the order is ...

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### Why do these two Monster-related calculations yield $163$?

Fact 1: (1979, Conway and Norton)$^{1}$
"There are $194-22-9=\color{blue}{163\,}$ $\mathbb{Z}$-independent McKay-Thompson series for the Monster."
Note: There are 194 (linear) irreducible ...

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### 71, the Monster, and c = 24 CFTs

The largest prime in the order of the Monster group is $71$. This number $71$ shows up at various places:
The minimal faithful representation has dimension $196883 = 47.59.71$
The Monster group can ...

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### Monstrous Langlands-McKay or what is bijection between conjugacy classes and irreducible representation for sporadic simple groups?

Context: The number of conjugacy classes equals to the number of irreducuble representations (over C) for any finite group. Moreover for the symmetric group and some other groups there is "good ...

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### The largest primes in the monster group construction

The last three prime numbers in the factorization of the order of the Fischer Griess friendly monster group are 47, 59, 71. (https://en.wikipedia.org/wiki/Monster_group)
On the other hand, the monster ...

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### Sporadic and Exceptional

I have been reading this recent paper of J.McKay and YH. He (they've written a number of papers recently, including a fun and joking one on 42 which overflow commented on) called "Sporadic and ...

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### A curious property of Ramanujan's function $\tau(n)$

As it is well known, Ramanujan's $\tau(n)$ function can be defined through the power series expansion of the modular discriminant:
$$\Delta(q)=q\prod\limits_{n=1}^\infty (1-q^n)^{24}=\sum \limits_{n=...

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### "A locally dual polar space for the Monster"

I am currently looking at Ronan and Stroth's 1984 paper Minimal Parabolic Geometries for the Sporadic Groups. When considering the $3$-minimal parabolic system of $F_{1}$, they cite a preprint by ...

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### Computing Thompson series for the monster group

I am trying to do some experimentation with the values of Thompson series, but I have having a hard time finding a table that has these Thompson series with as many terms as I'd like. The tables I've ...

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### $H^4$ of the Monster

The Monster group $M$ acts on the moonshine vertex algebra $V^\natural$.
Because $V^\natural$ is a holomorphic vertex algebra (i.e., it has a unique irreducible module), there is a corresponding ...

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### The Monster Group uses in mathematical physics

I am doing a project on the inverse Galois problem, and am seeking to show that the monster group is realisable over the rationals. I have heard that the monster group has found uses in theoretical ...

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### FLM-like construction of VOA for other simple groups

Frenkel, Lepowsky, Meurman constructed the vertex operator algebra (VOA) $V^\natural$ as a chiral orbifold CFT whose target space is $\mathbb{R}^{24}/\Lambda/\mathbb{Z}_2$. (Here the last $\mathbb{Z}...