Skip to main content

Questions tagged [sporadic-groups]

A sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups. All the other finite simple groups form 18 infinite families numbered by q - power of prime number and n - natural number. Sporadic groups attach attention due to their sporadic/exceptional nature - similar to exceptional Lie groups. The first sporadic groups were found by Mathieu in 1860s. The last sporadic group J4 was discovered in 1975 by Janko.

Filter by
Sorted by
Tagged with
6 votes
1 answer
364 views

Why do symmetries of K3 surfaces lie in the Mathieu group $M_{24}$?

I'm having trouble following some steps of this argument from the appendix of Eguchi, Ooguri and Tachikawa's paper Notes on the K3 surface and the Mathieu group M24: Now let us recall that the ...
John Baez's user avatar
  • 21.8k
2 votes
1 answer
178 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 ...
Daniel Sebald's user avatar
6 votes
1 answer
281 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 ...
Theo Johnson-Freyd's user avatar
1 vote
1 answer
98 views

On reducing degree-$12$ equations with Mathieu group $M_{11}$ to its degree-$11$ resolvent?

The general $4$-deg and some $8$-deg (such as the Schein octic) when a linear transformation is done so their $x^{n-1}$ term vanishes can have a neat solution as, $$x = \sqrt{z_1}+\sqrt{z_2}+\sqrt{z_3}...
Tito Piezas III's user avatar
8 votes
1 answer
459 views

Small subgroups of the monster

Is every group of order at most 36 isomorphic to a subgroup of the monster group?
Daniel Sebald's user avatar
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 ...
Daniel Sebald's user avatar
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 $M_{13}$....
Daniel Sebald's user avatar
2 votes
1 answer
156 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?
Daniel Sebald's user avatar
-4 votes
1 answer
141 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?
Daniel Sebald's user avatar
4 votes
3 answers
315 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 ...
John McVey's user avatar
  • 1,058
6 votes
1 answer
549 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 ...
OzoneNerd's user avatar
  • 179
10 votes
0 answers
473 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 ...
Adam P. Goucher's user avatar
20 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 ...
JamesEadon's user avatar
2 votes
2 answers
409 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 ...
JamesEadon's user avatar
1 vote
1 answer
421 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 ...
Sebastien Palcoux's user avatar
3 votes
0 answers
126 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 ...
Theo Johnson-Freyd's user avatar
3 votes
1 answer
214 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 ...
Andrei Smolensky's user avatar
4 votes
0 answers
236 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 ...
Theo Johnson-Freyd's user avatar
4 votes
1 answer
618 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 ...
wonderich's user avatar
  • 10.4k
19 votes
1 answer
414 views

Sporadic subgroup of E7

The dimensions of some representations of the Janko group J1 coinside with dimensions of smallest representations of the Lie algebra of type E7 (56, 133). It seems to be natural that there is a ...
Ievgen Makedonskyi's user avatar
14 votes
0 answers
707 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-...
user avatar
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: ...
Alexander Chervov's user avatar
5 votes
0 answers
533 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 ...
Tensor_Product's user avatar
9 votes
2 answers
522 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 ...
John Murray's user avatar
  • 1,070
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 ...
Colin Reid's user avatar
  • 4,688
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 ...
Simon Rose's user avatar
  • 6,240
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 ...
Simon Lentner's user avatar
5 votes
1 answer
368 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 $E_8$ ...
user avatar
7 votes
1 answer
413 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 ...
Thomas Connor's user avatar
54 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 ...
REDace0's user avatar
  • 677