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31
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0answers
1k views

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 $S_n$. ...
21
votes
0answers
713 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)\, ...
21
votes
0answers
348 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 ...
19
votes
0answers
242 views

Uniform proof that a finite reflection group is determined by its degrees?

Given a finite (real) group $G$ generated by reflections acting on euclidean $n$-space, it was shown by Chevalley in the 1950s that the algebra of invariants of $G$ in the associated polynomial ...
19
votes
0answers
528 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 ...
18
votes
0answers
671 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 ...
16
votes
0answers
335 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 ...
16
votes
0answers
414 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 ...
14
votes
0answers
228 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 ...
13
votes
0answers
449 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, ...
13
votes
0answers
506 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 ...
12
votes
0answers
531 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 ...
10
votes
0answers
218 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, ...
9
votes
0answers
207 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 ...
9
votes
0answers
214 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 ...
9
votes
0answers
399 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 ...
7
votes
0answers
171 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 ...
7
votes
0answers
248 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, ...
7
votes
0answers
110 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 ...
7
votes
0answers
218 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 ...
7
votes
0answers
312 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 ...
6
votes
0answers
228 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 ...
6
votes
0answers
162 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
votes
0answers
197 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 ...
6
votes
0answers
233 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 ...
6
votes
0answers
342 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 ...
6
votes
0answers
146 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 ...
6
votes
0answers
211 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 ...
6
votes
0answers
381 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 ...
6
votes
0answers
484 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 $(G, H)$ ...
5
votes
0answers
77 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 ...
5
votes
0answers
202 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 ...
5
votes
0answers
105 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 ...
5
votes
0answers
251 views

Vanishing sums of powers modulo n

Is it possible to describe all positive integer sequences $\{x_i\}$ such that $$ \sum_i x_i=n \quad \hbox{and} \quad \sum_i x_i^k\equiv 0\pmod n \quad \hbox{for all $k$ (and a given $n$)?} $$ ...
5
votes
0answers
138 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 ...
5
votes
0answers
335 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 ...
4
votes
0answers
75 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 ...
4
votes
0answers
309 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 ...
4
votes
0answers
243 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?
4
votes
0answers
130 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 ...
4
votes
0answers
184 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 ...
4
votes
0answers
243 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.
4
votes
0answers
175 views

Arithmetic relations between degrees of irrchar and cardinals of conjugacy classes

Perhaps this question arose already in MO. If so, then I'm ready to delete it. If $G$ is a finite group, I denote c $=(c_1,\ldots,c_r)$ the cardinals of its congacy classes and m $=m_1,\ldots,m_s$ ...
3
votes
0answers
168 views

Profinite groups, completions, and Schreier's formula

Let $G$ be a finitely generated profinite group, and $H \leq_o G$. We say that $H$ satisfies Schreier's formula in $G$ if $d(H) - 1 = (d(G)-1)[G:H]$. We say that $G$ satisfies Schreier's formula if ...
3
votes
0answers
90 views

Automorphisms of profinite groups

Let $d,n \in \mathbb{N}$, and $p$ a prime number. Let $F$ be a free pro-$p$ group on $d$ generators. Is there an automorphism of $F$ of order $n$?
3
votes
0answers
73 views

Double coset relation for unique intermediate subgroup (with homogeneity)

Let $G$ be a group and $H$ a subgroup such that there is a unique (non-trivial) intermediate subgroup $K$ (i.e. $H < S < G$ implies $S=K$) and $(H \subset K) \sim (K \subset G)$ (homogeneity) ...
3
votes
0answers
232 views

On the Groups of Order $(p^2+1)/2$

A few days ago I asked a question (Groups of order $p(p^2+1)/2$) about a finite group of order $p(p^2+1)/2$ and I got a lot of useful information about it. Thanks for the nice and very helpful ...
3
votes
0answers
85 views

On divisors occurring as subgroup sizes

Given a finite group $G$ define $D(G)$ to be the number of divisors $r$ of $|G|$ for which there exists a subgroup of $G$ of order $r$. Clearly $D(G) \leq d(|G|)$, where $d(n)$ denotes the number of ...
3
votes
0answers
234 views

How do I determine the smallest dimension of an irreducible $\mathbb{F}_p[G]$-module with a prescribed trivial fixed point space?

This is a crosspost from MSE since I haven't found an answer there yet. I am not very familiar with modular representation theory or Brauer theory yet, however lately I have needed to use ...
3
votes
0answers
193 views

How to find the normalizer of a finite subroup in a Lie group?

If a group $G$ is generated by finitely many subgroups $G_i$ and $H$ a subgroup of $G$, under which conditions can $N_G(K)$, the normalizer of $K$ in $G$, be generated by all the normailizers ...