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2
votes
0answers
239 views

Are the homogeneous single chain subfactors, Dedekind?

Background: See here and there. Recall that a subfactor is Dedekind if all its intermediate subfactors are normal. A subfactor $(N \subset M)$ is Homogeneous Single Chain (HSC) if its lattice ...
11
votes
1answer
491 views

Find finite groups $G\cong Aut(G)$

I become interested in this problem because $G\cong Aut(G)$ suggests a special symmetry in $G$. From $G/Z(G)\cong Inn(G)$ we know complete group is the anewer for the simplest case, though this class ...
4
votes
0answers
78 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 ...
1
vote
1answer
197 views

On the character degrees of a finite group with special structure

Let $G$ be a finite group such that $G$ has a normal subgroup $N$ of order $p(p^2+1)/2$, where $p>13$ is an odd prime and $p\ne 239$. Also $G/N\cong \text{PSL}(2,p)$. Can we say that there exists a ...
0
votes
0answers
88 views

Brauer characters of finite simple group $E_8(5)$

I would like to find the irreducible characters of the group $E_8(5)$ (mod 2)? Can anyone help? (I am elementary in working with Brauer characters) Many thanks
1
vote
1answer
185 views

List of Conjugacy Classes of the General Linear group over $\mathbb{Z}/p^2\mathbb{Z}$

Does anybody know the answer to this or a good way to go about working this out? I have a list for $GL_2(Z/pZ)$ and I am trying to lift it to this; I have mostly been using fairly elementary algebraic ...
-5
votes
1answer
190 views

on the solvable groups of order $p^aq^b$ [closed]

We know that if $ p$ is a prime number then $ O^p (G) $ is the smallest normal subgroup of $ G $ such that $ G/O^p (G) $ is a $ p $-group. Now let $ G $ be a finite group of order $ p^aq^b $ where $ ...
2
votes
1answer
211 views

In which finite groups is there a non-central g such that, for all irreducible characters, Chi(i)(g) <> zero?

What is the character of Pi(G), the tensor product of all inequivalent irreducible representations of G?
2
votes
0answers
197 views

The tallest possible lattice?

Let O be a complete discrete valuation ring and G a finite group. Recall that a finitely generated O-free OG-module $M$ such that the traces of the invertible endomorphisms of $M$ generate a strictly ...
5
votes
1answer
356 views

Finite groups for which the element orders form an arithmetic progression

Which are the finite groups $G$ such that the element orders of $G$ form an arithmetic progression? Several remarks: $S_3$, $A_4$ and any $p$-group of exponent $p$ satisfy this property. If $G$ ...
6
votes
1answer
275 views

The best upper bound for the number of involutions in a finite non-abelian simple group

Let $G$ be a finite non-abelian simple group and $t$ is equal to the number of involutions of $G$. We know that $t<|G|/3$ or $3t+1 \leq |G|$. Is this the best upper bound for the number of ...
4
votes
0answers
318 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 ...
2
votes
1answer
347 views

A second isomorphism theorem for the inclusions of groups

The usual second isomorphism theorem for groups is: let $G$ be a group, $S$ and $N$ subgroups with $N$ normal, then $SN$ is a subgroup of $G$, $S\cap N$ is a normal subgroup of $S$ and $SN/N \simeq ...
2
votes
1answer
213 views

on the prime divisors of $(p^2+1)/2 $

The following question is equivalent to a problem in group theory. Let $ p > 13$ be a prime number distinct from 239. Let $ a=(p^2+1)/2 $. Is there any prime divisor $r$ of $a$ such that $r\mid ...
3
votes
1answer
209 views

Normal intermediate subgroup and normal core

Let $G$ be a finite group and $H$ a subgroup. The normal core of $H$ in $G$ is $core_G(H) := \bigcap_{g \in G}g^{-1}Hg$ Definition: $K$ is a normal intermediate subgroup of the inclusion $(H ...
3
votes
0answers
235 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 ...
5
votes
5answers
914 views

Groups of order $p(p^2+1)/2$

It seems that when $p>3$ is a prime, then each group of order $p(p^2+1)/2$ is abelian as I checked by Gap for small $p$. Is it true for each $p$? Thanks for your answers
6
votes
5answers
454 views

Reference requested: Random walk on groups

I am looking for a good reference to learn about random walks on groups (either finite groups or Lie groups). Ideally, I would like a reference for general theory of random walks on groups that is ...
21
votes
0answers
743 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)\, ...
1
vote
0answers
169 views

$\mathcal{L}(H_i \subset G_i)$ distributive $\Rightarrow$ $\mathcal{L}(H_1 \times H_2 \subset G_1 \times G_2)$ modular?

Let $\mathcal{L}( G)$ be the lattice of subgroups of $G$ and $\mathcal{L}(H \subset G)$ the lattice of intermediate subgroups. Definitions: A lattice $(L, \wedge, \vee)$ is distributive if, ...
0
votes
2answers
138 views

Products of maximal inclusions of finite groups with a non-obvious intermediate

Let $(H_1 \subset G_1)$ and $(H_2 \subset G_2)$ be core-free maximal inclusions of finite groups. Their product, the inclusion $(H_1 \times H_2 \subset G_1 \times G_2)$, admits four obvious ...
2
votes
0answers
173 views

Existence of inclusions of finite groups with a particular lattice property

Definition : Let $\sim$ be the equivalence relation on inclusions of finite groups, generated by : $(H \subset G) \sim (\phi(H) \subset \phi(G))$, with $ \phi: G \to L$ a finite group morphism and ...
9
votes
3answers
463 views

Classification of automorphism groups of groups of order $p^4$

For the purpose of classifying another algebraic structure which is parametrised by the choice of a group and of an automorphism I need the classification up to isomorphism of automorphism groups of ...
8
votes
2answers
235 views

3-cocycle representatives for the dihedral group $D_{2n}$?

I am looking for a reference for a complete list of 3-cocycle representatives for $H^3(D_{2n},\mathbb{C}^\times)$, where $$ D_{2n}=\langle a, b\mid a^2=b^2=(ab)^n=e\rangle $$ is the dihedral group of ...
7
votes
0answers
114 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 ...
4
votes
1answer
156 views

To whom is the internal characterization of $Q$-groups due?

A group is said to be a $Q$-group if the character of any complex representation is rational valued. A well-known internal characterization of $Q$-groups is the following: $G$ is a $Q$-group if ...
1
vote
1answer
107 views

Index of agemo subgroups in $p$-groups

Having a finite $p$-group $G$ ($p$ odd). we denote by $\Omega_1(G)$ the subgroup generated by all the elements of $G$ of order dividing $p$. Is there an example of such a group $G$, such that ...
11
votes
4answers
468 views

Iterated semi-direct products

Let $G$ be a finite group. Suppose that we can write $G= A \rtimes B$ and also $A = C \rtimes D$. Further suppose that C is normal in $G$ (not just in $A$). Then can we write $G = C \rtimes E$ where ...
4
votes
2answers
306 views

Factor subset of finite group

Let $G$ be a group of order $n$ and $d$ a positive divisor of $n$. Is it true that there exists a subset $A$ of $G$ with $d$ elements and a subset $B$ such that $G=AB$ and $|AB|=|A||B|$ ...
13
votes
1answer
344 views

Does O'Nan-Scott depend on CFSG?

My question is in the title. Some context: there are two versions of the O'Nan-Scott theorem. The first, weaker version, is due to O'Nan and Scott (independently) and gives the structure of the ...
3
votes
2answers
294 views

n-cocycles of finite abelian groups from cohomology group

Question: Given a generic finite abelian group $G=\mathbb{Z}_{N^{(1)}} \times \cdots \times \mathbb{Z}_{N^{(k)}}$. (1) What is the explicit forms of its cohomology group (see my definition) in a ...
6
votes
6answers
428 views

Almost uniquely generated groups

This is inspired by this question. Does there exist an infinite finitely generated group having (a) a unique (b) finitely many inclusion-minimal generating set(s) up to ...
4
votes
1answer
199 views

Doubly primitive groups with simple socle

The classification of doubly transitive groups with simple socle is known. A good account of such classification can be found for example in this paper: Cameron, Peter J. Finite permutation groups ...
18
votes
2answers
610 views

Reference for the triple covering of A_6

I would like to ask for a reference (book, paper ...) for the following nice construction, which I have found as an exercise in some notes of a course by R. Borcherds. For $n=6$ or $7$ (and only in ...
4
votes
1answer
175 views

Cyclic subgroups of GL(n,q)

Let $q$ be a prime power. It is well known that every Singer subgroup (subgroup of order $q^n-1$) in $GL(n,q)$ are conjugate. My question is: If $H$ is a cyclic subgroup of order $m$ in $GL(n,q)$, ...
25
votes
2answers
892 views

A group-theoretic perspective on Frankl's union closed problem

Here is a group theoretic phrasing of a special case of the union closed conjecture: Question: Given a finite group $G$, is there an element of prime power order which is contained in at most half ...
5
votes
2answers
348 views

Decomposing the conjugacy representation of Sym$(n)$ for small $n$

I am trying to compute the decomposition of the conjugacy representation of some small symmetric groups. Perhaps someone has undertaken a similar calculation. My own calculations are quite slow, ...
11
votes
1answer
253 views

Group cochains invariant under the action of the symmetric group

Let $G$ be a finite group and $A$ an abelian group. Recall the cochain groups $$ C^k = \{f: G^k \to A\} $$ and the coboundary map $$ \delta : C^k \to C^{k+1} $$ $$ (\delta f)(g_1, \ldots, g_{k+1}) ...
0
votes
1answer
151 views

relatively free groups in $Var(S_3)$

Suppose $S_3$ is the symmetric group of order 6. Which elements of the variety $Var(S_3)$ are relatively free? This question is related to my previous question Relatively free algebras in a variety ...
16
votes
1answer
688 views

Examples of finite groups with “good” bijection(s) between conjugacy classes and irreducible representations?

For symmetric group conjugacy classes and irreducible representation both are parametrized by Young diagramms, so there is a kind of "good" bijection between the two sets. For general finite groups ...
4
votes
4answers
525 views

Structure of the adjoint representation of a (finite) group (Hopf algebra) ?

Every group acts on itself by conjugation $h \mapsto g h g^{-1}$. Respectively considering functions on a group we obtain a linear representation. Question 1: what is known about this representation ...
6
votes
2answers
265 views

Decomposing representations of GL(n,F_q) induced from certain kinds of parabolics

The answer to the question below is almost certainly known to the representation theorists; in fact, I'm pretty sure it can be extracted from Green's paper "The characters of the finite general linear ...
7
votes
2answers
361 views

How many finite loops?

How many finite loops of order $n$ are there? I am interested in the exact values ​​of $n$ if $n <40$ or even reasonable estimates. I am also interested in formulae or bounds for all $n$. Note ...
1
vote
1answer
208 views

The compact Lie group contains a finite subgroup $\mathbb{Z}_{n_1} \times \mathbb{Z}_{n_2} \times \mathbb{Z}_{n_3}$

Given a finite Abelian group: $G=\mathbb{Z}_{n_1} \times \mathbb{Z}_{n_2} \times \mathbb{Z}_{n_3}$, where ${n_1},{n_2},{n_3}$ are arbitrary positive integers. ${n_1},{n_2},{n_3}$ may have or may not ...
-4
votes
1answer
207 views

Simple group of order 504 [closed]

As we know,there are 9 Sylow 2-subgroup in the Simple group of order 504.Can anyone prove it only by Sylow's theorem? (you can't use knowledge about PSL(2,8))
15
votes
1answer
620 views

Are There Always Group Generators Which Give Unimodal Growth?

Suppose $G$ is a $k$-generated finite group. Is there always a set of $k$ elements which generate the group and have a unimodal counting function? Background: The counting function, $f(n)$, is a ...
2
votes
1answer
94 views

Flag primitivity of the correlation group of classical projective planes.

We know that the full automorphism group of the $\pi_q = PG(2,q)$ acts imprimitively on the flags (all flags through a fixed point form a block). But, things change when we consider the action of full ...
7
votes
2answers
516 views

Number of isomorphism types of finite groups

Are there some good asymptotic estimations for the number $F(n)$ of non-isomorphic finite groups of size smaller than $n$?
3
votes
2answers
312 views

Quotient of a rational variety by a finite group

Let $X$ be a rational variety and let $G$ be a finite group acting on $X$. Let us consider the diagonal action of $G$ over the product $X^{h} = X\times...\times X$, $$G\times(X\times...\times ...
9
votes
0answers
413 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 ...