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### Normalizers in symmetric groups

Question: Let $G$ be a finite group. Is it true that there is a subgroup $U$ inside some symmetric group $S_n$, such that $N(U)/U$ is isomorphic to $G$? Here $N(U)$ is the normalizer of $U$ in $S_n$.
...

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436 views

### Class function counting solutions of equation in finite group: when is it a virtual character?

Let $w=w(x_1,\dots,x_n)$ be a word in a free group of rank $n$. Let $G$ be a finite group. Then we may define a class function $f=f_w$ of $G$ by
$$ f_w(g) = |\{ (x_1,\dots, x_n)\in G^n\mid ...

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774 views

### Monstrous Moonshine for Thompson group $Th$?

I. As a background, in Traces of Singular Moduli (p.2), Zagier defines the modular form of weight 3/2,
$$g(\tau) = \frac{\eta^2(\tau)}{\eta(2\tau)}\frac{E_4(4\tau)}{\eta^6(4\tau)}=\vartheta_4(\tau)\, ...

**19**

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557 views

### Is there a Mathieu groupoid M_31?

I have read something which said that the large amount of common structure between the simple groups $SL(3,3)$ and $M_{11}$ indicated to Conway the possibility that the Mathieu groupoid $M_{13}$ might ...

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429 views

### Density of first-order definable sets in a directed union of finite groups

This is a generalization of the following question by John Wiltshire-Gordon.
Consider an inductive family of finite groups:
$$
G_0 \hookrightarrow G_1 \hookrightarrow \ldots \hookrightarrow G_i ...

**18**

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727 views

### Given a lattice L with n elements, are there finite groups H < G such that L $\cong$ the lattice of subgroups between H and G?

If there is no restriction on $n$, this is a famous open problem. I'm wondering if any recent work has been done for small $n>6$. I believe the question is answered (positively) for $n=6$ by ...

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358 views

### Applications of the surjectivity of Brauer's decomposition map over arbitrary fields?

Recently I've been going over some of Serre's reformulation of Brauer theory with a student, following the influential treatment in Part III of Serre's lectures (revised 1971 French edition) later ...

**15**

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280 views

### Maximum automorphism group for a 3-connected cubic graph

The following arose as a side issue in a project on graph reconstruction.
Problem: Let $a(n)$ be the greatest order of the automorphism group of a 3-connected cubic graph with $n$ vertices. Find a ...

**14**

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515 views

### What's the big deal about $M_{13}$?

$M_{13}$ is the Mathieu groupoid defined by Conway in
Conway, J. H. $M_{13}$. Surveys in combinatorics, 1997 (London), 1–11,
London Math. Soc. Lecture Note Ser., 241, Cambridge Univ. Press, ...

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540 views

### How much has been written down about Deligne's geometric approach to the order formula for a finite group of Lie type?

This is a follow-up to a recent mathoverflow question
34387
about computing the orders of finite unitary groups and the comments made there.
Between 1955 (Chevalley's Tohoku paper) and 1968 ...

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589 views

### What can be said about Schur indices, given only the character table?

Let $\chi$ be an irreducible (complex) character of a finite group, $G$. The Schur index $m_{K}(\chi)$ of $\chi$ over the field $K$ is the smallest positive integer $m$ such that $m\chi$ is afforded ...

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176 views

### Elements of order 3 normalizing no non-identity 2-subgroups in Almost Simple Groups

This question is partly motivated by a situation which arises in modular representation theory. A finite group $G$ is said to be almost simple if $G$ has a unique minimal normal subgroup which is a ...

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284 views

### Generating random finite groups

I would like a method to efficiently generate a random finite group of a given order $n$.
If there are $g(n)$ non-isomorphic groups of order $n$,
ideally each group would occur with probability ...

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314 views

### $2$-group with two isomorphic normal subgroups of index $4$ with non-isomorphic quotients

Hypothesis: Let $P$ be a finite $2$-group with two isomorphic normal subgroups $M$ and $N$ such that $P/M\cong C_4$ (the cyclic group of order $4$) while $P/N\cong C_2^2$. By the lattice theorem, ...

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184 views

### Solving a set of equations in a finite symmetric group

A standard way to find solutions to a finite set of equations in a finite symmetric group
${\rm S}_n$ is to take the equations as relators of a finitely presented group, to use
the low index subgroups ...

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266 views

### Non split extension isomorphic (as a group) to a split extension

$\def\Z{\mathbb{Z}}$
Let $A$ be a finite abelian group and $G$ a finite group acting on $A$.
Then the extension $0 \to A \to E \to G \to 1$ is splits if and only if the corresponding $2$-cocycle is ...

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423 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 ...

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190 views

### A strong sum-product “for translates” in finite fields

In the course of some recent research, I've sketched out a proof of the following result. My basis question is: is the result interesting?
Proposition There exists an absolute constant $c$ such ...

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260 views

### Connection between two theorems on character values?

In a recent arXiv preprint here, Dipendra Prasad has revisited a 1976 theorem of Kostant (Theorem 2 in the paper On Macdonald's $\eta$-function formula, the Laplacian and generalized exponents, ...

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121 views

### Fusion pattern in a cyclic subgroup of order 8

Can a finite simple group $G$ have an element $x$ of order 8 such that $x^2$ is conjugate to $x^{-2}$ but $x$ is not conjugate to any element of $\{x^3,x^5,x^7\}$?
In other words, does this fusion ...

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272 views

### Outer group automorphisms preserving conjugacy classes of pairs of commuting elements

The following problem appears in the study of braided autoequivalences of representation categories of Drinfeld doubles of finite groups.
Let $G$ be a finite group. An outer automorphism $\alpha$ of ...

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420 views

### Cyclic Sylow $p$-subgroup in finite simple groups

Hello,
I came accross a beautiful and "simple" (no pun intended) theorem, mentioned here in slide 14 by Jack Schmidt.
All finite simple groups have a cyclic
Sylow $p$-subgroup for some $p$
I ...

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234 views

### Why would dim primitive irrep divide size of some conjugacy class ?

From Isaacs et.al. 2005
Conjecture C. Let χ be a primitive
irreducible character of an arbitrary
ﬁnite group G. Then χ(1) divides |
clG(g)| for some element g ∈ G.
Here, of course, we ...

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332 views

### The maximal order of an element in orthogonal groups over finite fields of characteristic 2

Let $q$ be a power of $2$ and let $(V,Q)$ be a
quadratic space of dimension $2m$ over $\mathbb{F}_q$. Up to isometry, we know that we have exactly two classes of such quadratic spaces: the plus type ...

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174 views

### A constant associated to the character table of a finite group

Following on from some of myprevious MO questions on finite group theory...
$\newcommand{\Irr}{\operatorname{Irr}}\newcommand{\Conj}{\operatorname{Conj}}\newcommand{\AMZL}{{\rm AM}_{\rm Z}}$
Let $G$ ...

**6**

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212 views

### Intersections of anisotropic tori with split Levi subgroups

Let G be a connected reductive group defined and split over a finite field k with Frobenius morphism F. Let T be an F-stable minisotropic maximal torus. Let P be an F-stable proper parabolic ...

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237 views

### Recovering a group from its number of symmetric embeddings?

Fix a finite group $G$, and let $f_G:\mathbb{N}\rightarrow\mathbb{N}$ be defined by setting $f_G(n)$ to be the largest $m$ such that $S_n$ contains $m$ disjoint (pairwise-intersecting at 1) copies of ...

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149 views

### Finding a database of representations as matrices

Sorry if this would be more appropriate as a stackoverflow and not a mathoverflow question, but I think it's more likely to be known in this community.
There are plenty of places on the internet or ...

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213 views

### Cohomology of T^{n}/W for compact Lie groups

Let $G$ be a compact, connected and simply connected.
Let $T\subset G$ be a maximal torus and let $W$ be the corresponding
Weyl group.
Then we have the diagonal action of $W$ on $T^{n}$ for $n\ge ...

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412 views

### Jordan's Theorem on primitive permutation groups

The theorem I am referring to in the title is this:
Theorem. If $p$ is a prime and $n$ is an integer with $n \geq p+3$, then the only primitive permutation groups on $n$ points containing a ...

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491 views

### When is Hom(G, H) the same size as Hom(H, G)?

Let $G$ and $H$ be finite groups. Consider the ratio
$$r_{G, H} \equiv {|Hom(G, H)| \over{|Hom(H,G)|}}$$
My question is
When is $r_{G, H} = 1$? Can we characterize the pairs of groups $(G, H)$ ...

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149 views

### Derived subgroup of rational points versus rational points of derived subgroup

Let $\mathbf G$ be a connected algebraic group defined over a field $\mathbb F_p$. If $q=p^n$, then the groups $\mathbf G^\prime (\mathbb F_q)$ and $\mathbf G (\mathbb F_q)^\prime$ are not always ...

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162 views

### Approximating Lie groups by finite groups

How can one approximate compact Lie groups by finite groups?
My wish is something like this:
Let $G$ be a compact Lie group.
There is a sequence of nested finite subgroups $G_n$ so that $G_n\to ...

**5**

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82 views

### Uniqueness of the direct product decomposition of inclusions of finite groups

This post is a generalization of Uniqueness of the direct product decomposition of finite groups.
Here we look inclusions of finite groups $(H \subset G)$ instead of just finite groups.
Definition: ...

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113 views

### Self-duality of the subgroup lattice of $G\times H$

Let $G$ and $H$ be finite groups and $\gcd(|G|,|H|)=1$.
Suppose the lattice of all subgroups of $G\times H$ is self-dual. Is the lattice of all subgroups of $G$ self-dual?

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84 views

### representable functions on FinGrp

We look at functions $f: \text{FinGrp} \rightarrow \mathbb{N}$ such that $f$ is constant on isomorphism classes.
Let's say that $f$ is representable if there is a (possibly infinite) group $K$ such ...

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217 views

### A particular example of solvable group

Let be $m$ an integer, $m>0$, $n=p_1^{a_1}...p_t^{a_t}$. Define $f(n)=a_1+...+a_t$.
Let $G$ be a finite group and define $f(G)=\max\left\{\,f(\lvert g\rvert):g\in G\,\right\}$.
Is there a finite ...

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111 views

### Information about permutation character from local action

Let $G$ be a finite permutation group acting transitively, but not regularly, on a set $V$. Let $H$ be the stabilizer of some point $v\in V$, and suppose that $H$ acts 2-transitively on one of its ...

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257 views

### Vanishing sums of powers modulo n

Is it possible to describe all positive integer sequences $\{x_i\}$ such that
$$
\sum_i x_i=n
\quad
\hbox{and}
\quad
\sum_i x_i^k\equiv 0\pmod n
\quad
\hbox{for all $k$ (and a given $n$)?}
$$
...

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144 views

### Subgroups of M-groups

Let $G$ be an M-group. Let $\chi$ be an irreducible character of $G$ and set $$A_{\chi} =
\{ H\leq G\ |\ \exists \psi\in\textrm{Lin}(H): \psi^G = \chi\}$$
Since $G$ is an M-group, this set is ...

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338 views

### What are the relation between Rep(G) and Rep(S_n)?

Let G be a finite group. We know it can be written as a subgroup of S_n. On the other hand, people sometimes say Rep(G) --- the category of all finite dimensional representations, are more interesting ...

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92 views

### Improvements of the Reidemeister-Schreier index formula for particular classes of groups

I have a couple of questions regarding possible improvements of the Reidemeister-Schreier index formula: let $G$ be a $d$-generated group and let $H$ be a subgroup of $G$, then
$$d(H) \le (d-1) ...

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187 views

### In general, are 'Young symmetrisers' given by Littlewood-Richardson 'Orthogonal projection Operators'?

Consider $V^{\otimes n}$ where $V$ is vector space and the representation of GL(V) acting in the usual way. Now if I consider tensor products or plethysms of irreducible spaces, this is not in general ...

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145 views

### Is there a nontrivial profinite word which is trivial in any group with at most d generators?

Let $F$ be a free profinite group of rank $\aleph_0$, and let $d \in \mathbb{N}$.
Let $N_d \lhd_c F$ be the intersection of all open normal subgroups $L \lhd_o F$ for which $F/L$ can be generated by ...

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79 views

### Automorphisms with an orbit “transversal” to a subgroup

Let $G$ be a finite group, $H$ be a subgroup of index $2$, $\phi\colon G\to G$ an automorphism. For $d\in \mathbb N$ let us say that $\phi$ is $d$-transversal to $H$ iff there is $h\in H$ such that
...

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325 views

### More on Moonshine for the Thompson group and weakly holomorphic weight one half modular forms

This question is a follow-up to
Monstrous Moonshine for Thompson group $Th$?
and is based on various comments to that question, in particular S. Carnahan's mention of the
connection to known ...

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**0**answers

248 views

### nilpotent fixed-point-free groups of automorphisms

Let $G$ be a finite group and $H$ be a nilpotent subgroup of
$Aut(G)$. If $C_{G}(H)=1$, is $G$ solvable?

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135 views

### What is known about 2-modular representations of Ree groups of type $F_4$?

A too-vaguely worded question posted today about Suzuki and Ree groups reminds me to revisit a concern I never followed up years ago when assembling information about modular representations of finite ...

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201 views

### Interplay between two definitions of the transfer homomorphism.

The transfer homomorphism can be defined in a couple of ways. For the purposes of this question, assume that all groups are finite.
Defintion 1. Let $1\leqslant K'\leqslant L \leqslant K\leqslant ...

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244 views

### Can one characterize a permutation character from properties of the character table?

Sufficient conditions would establish the existence of a subgroup
stabilizing a point in the permutation representation.