All Questions
Tagged with sporadic-groups gr.group-theory
28 questions
14
votes
2
answers
742
views
Solving the Bring quintic using the Monster?
I. Method
Hermite's method to solve the Bring quintic by functions that obey $x^8+y^8=1$ implicitly uses octahedral symmetry, while Emil Jann Fiedler's solution by the Rogers-Ramanujan continued ...
- 12.6k
2
votes
1
answer
197
views
$\mathrm{PSL}_3(4)$ inside the Monster group
Which quasisimple groups with central quotient $G\cong\mathrm{PSL}_3(4)$ are isomorphic to subgroups of the Monster sporadic group? So far I know that $G$ itself is not and that $2\cdot G$, $2^2\cdot ...
- 2,773
6
votes
1
answer
288
views
Which finite simple groups are rational-relative-real?
A finite group $G$ is called rational if every element $g \in G$ is conjugate to all of its primitive powers $g^a, a \in (\mathbb{Z}/\operatorname{order}(g))^\times$.
Analogously, I'll call $G$ real ...
- 54.6k
8
votes
1
answer
485
views
Small subgroups of the monster
Is every group of order at most 36 isomorphic to a subgroup of the monster group?
- 2,773
3
votes
0
answers
123
views
$2^2 \cdot U_6(2)$ and $2^2.2^{1+20}U_6(2)$ in $\mathbb{M}$
In the first diagram of this paper, there are conjugacy classes of subgroups of the Monster group which are labeled $2^2 \cdot U_6(2)$ and $2^2.2^{1+20}U_6(2)$, respectively. Can subgroups in the ...
- 2,773
5
votes
0
answers
339
views
Does $\mathit{Suz}$ contain $M_{13}$?
$\newcommand\Suz{\mathit{Suz}}$I recently noticed that the Suzuki group $\Suz$ has as subgroups classes of both $L_3(3)$ and $M_{12}$, both of which are also subgroups of the Mathieu groupoid $M_{13}$....
- 2,773
2
votes
1
answer
157
views
Fusing the $\mathrm{PGL}(2,11)$ conjugacy classes of $\mathrm{Aut}(M_{12})$
Is there an embedding of $\mathrm{Aut}(M_{12})$ into the automorphism group of some larger sporadic group that fuses its two conjugacy classes of $\mathrm{PGL}(2,11)$ subgroups?
- 2,773
-4
votes
1
answer
143
views
Conjugacy classes of $PSL_2(11)$ and $PGL_2(11)$ in $Aut(HN)$
How many conjugacy classes each of $PSL_2(11)$ and $PGL_2(11)$ subgroups are contained in the automorphism group of the Harada-Norton group?
- 2,773
4
votes
3
answers
328
views
Is a point stabilizer in the Mathieu group $M_{20}$ half-transitive?
The background: We recall/define the following:
$\Omega_n=\{1,\dots,n\}$.
$M_n$ is the Mathieu group of degree $n$. We follow the Wikipedia article "Mathieu group" and define these groups ...
- 1,068
6
votes
1
answer
587
views
What are the "simplest" polytopes with an automorphism group of $\mathrm M_{11} \hspace {-1.25pt} $?
Do any polytopes have an automorphism group of the smallest of the sporadic groups, the Matthieu group $\mathrm M_{11} \hspace {-1pt} $? Indeed, they must exist. What are the simplest such polytopes ...
- 179
10
votes
0
answers
494
views
A lattice with Monster group symmetries
The book Mathematical Evolutions contains the following excerpt:
A last, famous, example is the following. It is known that in the space
of one hundred and ninety six thousand eight hundred and ...
- 12.4k
21
votes
3
answers
4k
views
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 ...
- 373
2
votes
2
answers
419
views
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 ...
- 373
1
vote
1
answer
431
views
The sporadic numbers
Let call $n$ a sporadic number if the set of groups $G \neq A_n,S_n$ having a core-free maximal subgroup of index $n$ is non-empty and contains only sporadic simple groups.
By GAP, the set of all the ...
3
votes
0
answers
127
views
Are all exceptional Schur covers sub-sporadic?
Famously, all but finitely many finite simple groups are (cyclic or alternating or) of Lie type. The groups of Lie type have central extensions coming from the simply connected covers of the ...
- 54.6k
3
votes
1
answer
223
views
Fixed points of the automorphisms of sporadic groups
Sporadic groups have very few outer automorphisms (in fact, $|\mathrm{Out}(G)|\leqslant2$), so it is very natural to ask what are the fixed points subgroups. For a group of Lie type (and a suitable ...
- 3,186
4
votes
0
answers
250
views
Normalizers of abelian Sylows in simple groups
Suppose $G$ is a (nonabelian) finite simple group and $p$ is a prime such that the $p$-Sylow in $G$ is abelian. What can be said about its normalizer? I'm particularly interested in lower bounds on ...
- 54.6k
4
votes
1
answer
632
views
Schreier conjecture -- without a simple proof? and sporadic simple groups
The Schreier conjecture asserts that $\mathrm{Out}(G)$ is always a solvable group when $G$ is a finite simple group. This result is known to be true as a corollary of the classification of finite ...
- 10.5k
14
votes
0
answers
716
views
Algebra for the Baby
I am reading the following article.
Ryba, Alexander J.E., A natural invariant algebra for the Baby Monster group., J. Group Theory 10, No. 1, 55-69 (2007). ZBL1228.20012..
Author works with 4370-...
user21230
14
votes
3
answers
1k
views
Construction of representations of the Mathieu groups?
The Mathieu groups are beautiful simple finite groups. (They were the first sporadic groups to be discovered in 1861-1870).
They are related with many other miraculous constructions in mathematics:
...
- 24.8k
5
votes
0
answers
558
views
Atlas of finite groups, Character table of automorphism group of sporadic group
I am consulting ATLAS of finite group for character table of Automorphism Group of sporadic group.
I am reading from Inverse Galois Theory by G. Malle
Let me start with $G=M_{12}$
This(image ...
- 591
9
votes
2
answers
526
views
Strongly real elements of odd order in sporadic finite simple groups
Recall that an element of a finite group is said to be real if it is conjugate to its inverse, and strongly real if the conjugating element can be chosen to be an involution.
Question: Is it true ...
- 1,090
35
votes
2
answers
1k
views
Why do sporadic simple groups have so few conjugacy classes?
In finite group theory, there's a general intuition that the further away a group is from abelian, the fewer conjugacy classes it will have. So it is to be expected that non-abelian finite simple ...
- 4,728
16
votes
0
answers
1k
views
How many sporadic simple groups are there, really?
I attended a talk by John Conway recently where he explained to us that the usual number, 26, was wrong, that there are in fact 27 sporadic simple groups. His reason was that the Tits group, which is ...
- 6,290
20
votes
2
answers
1k
views
(weak?) BN-Pair / Tits System for Sporadic Groups
The structure of finite simple groups of Lie type of arbitrary rank can be described well via BN-pairs. BN-pairs basically generalize the Bruhat decomposition of matrices into monomial $N$ and ...
- 1,846
5
votes
1
answer
371
views
Geometric interpretation of $2A$ conjugacy class in Conway group $Co_1$
I am struggling with following problem. Consider $2A$ class in $Co_1$ having $819*759*75$ elements. Each element $a$ from $2A$ have two representatives in $Co_0$. Element $a$ corresponds to $E_8$ ...
user21230
7
votes
1
answer
416
views
A rank 3 geometry for the sporadic simple group of Suzuki
I am actually studying coset geometries (in the sense of Tits and Buekenhout) for the sporadic simple group of Suzuki. I came aware that Buekenhout found in 1979 a geometry over the following diagram
...
- 499
55
votes
5
answers
10k
views
Why are the sporadic simple groups HUGE?
I'm merely a grad student right now, but I don't think an exploration of the sporadic groups is standard fare for graduate algebra, so I'd like to ask the experts on MO. I did a little reading on them ...
- 687