27
votes
Small subgroups of the monster
No, but almost.
$\newcommand\Dic{\mathit{Dic}}$It turns out that all isomorphism types of groups of order up to 36 occur as subgroups of the Monster with a single exception: no subgroup of the monster ...
- 5,654
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
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 ...
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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-...
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8
votes
Sporadic subgroup of E7
The Frobenius–Schur indicators of both $56$-dimensional representations are $+1$, meaning that in both cases the representation supports an invariant symmetric bilinear form. Since these are irreps, ...
- 54.6k
8
votes
Accepted
Fixed points of the automorphisms of sporadic groups
I think the information you are looking for is in Table~5.3 of GLS3:
Gorenstein, Daniel; Lyons, Richard; Solomon, Ronald.
The classification of the finite simple groups. Number 3. Part I.
...
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6
votes
Accepted
Which finite simple groups are rational-relative-real?
Rational-relative-to-real groups are frequently called inverse semi-rational groups (or cut groups). Inverse semi-rational groups are a subclass of semi-rational groups. A finite group $G$ is semi-...
- 133
6
votes
Accepted
On reducing degree-$12$ equations with Mathieu group $M_{11}$ to its degree-$11$ resolvent?
The answer is yes: $M_{11}$ in its action of degree $12$ has a subgroup of index $11$ (some people call it $M_{10}$) such that there is a $6$-element subset $A$ of the $12$ points which $M_{11}$ acts ...
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6
votes
Why do symmetries of K3 surfaces lie in the Mathieu group $M_{24}$?
This is clearly not the best reference on the subject — I would recommend Mukai's original paper: Finite groups of automorphisms of K3 surfaces and the Mathieu group, Invent. Math. 94 (1988), no. 1, ...
- 38k
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
6
votes
Accepted
Is a point stabilizer in the Mathieu group $M_{20}$ half-transitive?
No it is not: half-transitive means that all orbits have equal size (and the groups acts non-trivially), but this is not the case here. One can verify this e.g. using GAP:
...
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6
votes
Schreier conjecture -- without a simple proof? and sporadic simple groups
Regarding your second question, for every finite simple group the structure of its outer automorphism group is known.
For the sporadic groups, there is the following 2011 preprint on arXiv:
...
- 2,217
5
votes
Construction of representations of the Mathieu groups?
(This is not really an answer — or merely a very partial one —, but a clarification of the comment I posted earlier, as per OP's request.)
Consider the case of $M_{12}$. Let $V$ be its ($11$-...
- 32.5k
5
votes
The sporadic numbers
I cannot give a complete answer to this question right now, but I believe that it would be possible to answer it by writing a moderate amount of computer code that made use of existing results in the ...
- 37.4k
5
votes
$\mathrm{PSL}_3(4)$ inside the Monster group
The group $3.G$ centralizes an element of order three. If $3.G$ is a subgroup of the Monster then it is contained in a 3A centralizer (structure $3.Fi_{24}'$), a 3B centralizer (structure $3^{1+12}_+....
4
votes
Conjugacy classes of $PSL_2(11)$ and $PGL_2(11)$ in $Aut(HN)$
Two of ${\rm PSL}(2,11)$ (of lengths 103420800000 and 413683200000);
and two of ${\rm PGL}(2,11)$ (of lengths 206841600000 and 413683200000).
- 37.4k
4
votes
Is a point stabilizer in the Mathieu group $M_{20}$ half-transitive?
I certainly appreciate Max's answer (especially the geometric interpretation) and have accepted it as such. However, as is so usual, writing up the question caused me to focus on it with greater ...
- 1,068
4
votes
What are the "simplest" polytopes with an automorphism group of $\mathrm M_{11} \hspace {-1.25pt} $?
$\color {red} {\textbf {WARNING}}$
I believe @M.Winter has shown in the comments that the following answer is, in fact, $\color {red} {\textbf {incorrect}}$. It would appear that I have merely ...
- 179
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
4
votes
Strongly real elements of odd order in sporadic finite simple groups
One more confirmation of OP's observation can be found in tables,
from the paper:
Jordan Journal of Mathematics and Statisticscs (JJMS) l(2), 2008, pp. 97-103
97
STRONGLY REAL ELEMENTS IN
SPORADIC ...
- 24.9k
3
votes
Construction of representations of the Mathieu groups?
I would like to draw your attention to an article by Nick Gill and (his then MMath student) Sam Hughes The character table of a sharply 5-transitive subgroup of $A_{12}$, that constructs the character ...
- 3,451
2
votes
Construction of representations of the Mathieu groups?
Not really an answer to the question, but check out the very cool Bachelor's Thesis by A. Joshi (from IIT Madras).
- 96.4k
2
votes
Is a point stabilizer in the Mathieu group $M_{20}$ half-transitive?
To add to the previous answers, $M_{19}$ is the stabiliser of five points. In $M_{24}$, five points determine a unique octad containing them. The full setwise stabiliser of the octad is $2^4\!:\!A_8\...
- 16.2k
1
vote
Accepted
Solving the Bring quintic using the Monster?
(New answer):
One approach was to express the modular lambda function $\lambda(\tau)=A^8(\tau)$ in terms of the cubic continued fraction $C(\tau)$. For example, define,
\begin{align}
A(\tau) &= \...
- 12.6k
1
vote
Solving the Bring quintic using the Monster?
(A tentative answer using analogues of Hermite's approach.)
I've been analyzing the situation from Ramanujan's theories of elliptic functions to alternative bases of signature $(2,3,4,6)$. Hermite's ...
- 12.6k
1
vote
Fusing the $\mathrm{PGL}(2,11)$ conjugacy classes of $\mathrm{Aut}(M_{12})$
Norton and Wilson - Maximal subgroups of the Harada–Norton group seems to imply that $\mathrm {HN} \rtimes C_2$ works, along with its overgroups $\mathbb{B}$ and $\mathbb{M}$.
Edit:… or not. This ...
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