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11
votes
3answers
1k views

Is there a purely group-theoretic reformulation of an equivalence of subgroups?

There is an equivalence relation between inclusion of finite groups coming from the world of subfactors: Definition: $(H_{1} \subset G_{1}) \sim(H_{2} \subset G_{2})$ if $(R^{G_{1}} \subset ...
16
votes
1answer
608 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 ...
0
votes
2answers
125 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 ...
26
votes
2answers
1k views

Are there “real” vs. “quaternionic” conjugacy classes in finite groups?

The complex irreps of a finite group come in three types: self-dual by a symmetric form, self-dual by a symplectic form, and not self-dual at all. In the first two cases, the character is real-valued, ...
6
votes
2answers
453 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
1answer
196 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 ...
2
votes
0answers
220 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 ...
2
votes
1answer
326 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 ...
22
votes
7answers
4k views

The finite subgroups of SU(n)

This question is inspired by the recent question "The finite subgroups of SL(2,C)". While reading the answers there I remembered reading once that identifying the finite subgroups of SU(3) is still an ...
27
votes
4answers
2k views

For which $n$ is there only one group of order $n$?

Let $f(n)$ denote the number of (isomorphism classes of) groups of order $n$. A couple easy facts: If $n$ is not squarefree, then there are multiple abelian groups of order $n$. If $n \geq 4$ is ...
23
votes
9answers
2k views

How many groups of size at most n are there? What is the asymptotic growth rate? And what of rings, fields, graphs, partial orders, etc.?

Question. How many (isomorphism types of) finite groups of size at most n are there? What is the asymptotic growth rate? And the same question for rings, fields, graphs, partial orders, etc. ...
18
votes
8answers
2k views

Fantastic properties of Z/2Z

Recently I gave a lecture to master's students about some nice properties of the group with two elements $\mathbb{Z}/2\mathbb{Z}$. Typically, I wanted to present simple, natural situations where the ...
2
votes
2answers
289 views

If all real conjugacy classes are strongly real, then all real irreps are “strongly real”(symmetric), true ?

Question Is true that if all real conjugacy classes of a finite group are strongly real, then all its real irreducible representations (irreps) are "strongly real"(symmetric) ? And vice verse ? ...
4
votes
2answers
528 views

Decomposing representations of finite groups

Let $G$ be a finite group, $p$ a prime number. We denote by $\mathbb{F}_p$ the field of cardinality $p$. Let $V$ be an infinite dimensional representation of $G$ over $\mathbb{F}_p$. Must there be ...
5
votes
2answers
333 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, ...
4
votes
4answers
436 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 ...
2
votes
1answer
192 views

An upper bound for the maximal subgroups at fixed index?

Let us call a subgroup an injective homomorphism between groups. I warn the reader that a subgroup designates here an inclusion $(H \subset G)$, not $H$ alone. A subgroup $H \subset G$ is ...
0
votes
1answer
148 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 ...
18
votes
6answers
1k views

Measures of non-abelian-ness

Let $G$ be a finite non-abelian group of $n$ elements. I would like a measure that intuitively captures the extent to which $G$ is non-commutative. One easy measure is a count of the non-commutative ...
18
votes
1answer
1k views

Order of products of elements in symmetric groups

Let $n \in \mathbb{N}$. Is it true that for any $a, b, c \in \mathbb{N}$ satisfying $1 < a, b, c \leq n-2$ the symmetric group ${\rm S}_n$ has elements of order $a$ and $b$ whose product has order ...
16
votes
3answers
1k views

Which groups have only real and quaternionic irreducible representations?

Consider a continuous irreducible representation of a compact Lie group on a finite-dimensional complex Hilbert space. There are three mutually exclusive options: 1) it's not isomorphic to its dual ...
10
votes
2answers
512 views

A group action of the Heisenberg group with special symmetries

Suppose we look at the Heisenberg group $H_{d}$ as a matrix group of upper triangular matrices over the ring $\mathbb{Z}/d\mathbb{Z}$. You can even choose $d$ to be prime if you want. A natural ...
19
votes
4answers
2k views

Generating a finite group from elements in each conjugacy class

Is there a finite group such that, if you pick one element from each conjugacy class, these don't necessarily generate the entire group?
18
votes
2answers
683 views

Definition of “finite group of Lie type”?

The list of finite simple groups of Lie type has been understood for half a century, modulo some differences in notation (and identifications between some of the very small groups coming from ...
17
votes
3answers
874 views

Non-split extensions of $GL_n(F_q)$ by $F_q^n$ ?

A very naive question : I just learned that there is a non-split extension of $GL_3(F_2)$ by $F_2^3$ (with standard action). It can be realized as the subgroup of the automorphism group $G_2$ of ...
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
5answers
2k views

Generating finite simple groups with $2$ elements

Here is a very natural question: Q: Is it always possible to generate a finite simple group with only $2$ elements? In all the examples that I can think of the answer is yes. If the answer is ...
8
votes
3answers
1k views

Algorithm to check is representation irreducible ? Algorithm to decompose the reducible one ?

Question 1 Given a representation of a finite group, what algorithm can be used to check is it irreducible or not ? (Main case - complex numbers, comments on other cases are also welcome. "Given" ...
6
votes
3answers
1k views

Subgroups of p-groups

If $G$ is a (non-abelian) $p$-group, $|G|=p^n$, $n>3$, then it is elementary that $G$ contains a (normal) abelian subgroup of order $p^2$. It is also true that $G$ necessarily contains a normal ...
17
votes
3answers
873 views

Highly transitive groups (without assuming the classification of finite simple groups)

What is known about the classification of n-transitive group actions for n large without using the classification of finite simple groups? With the classification of finite simple groups a complete ...
12
votes
4answers
1k views

Has any attempt been made to classify finite groupoids?

I recently stumbled upon the Mathieu groupoid and I found them fascinating. It appears as a subset of $S_{13}$ which is not closed under multiplication, but it turns out to be a groupoid with 13 ...
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 ...
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)\, ...
8
votes
1answer
471 views

Periodic Automorphism Towers

In Scott's classic textbook on Group Theory, he asks: Suppose that $G$ is a finite group. Is the sequence of isomorphism types of the groups $Aut^{(n)}(G)$ for $n \in \mathbb{N}$ eventually periodic? ...
1
vote
1answer
174 views

smooth quotient out of a singular variety?

If $X$ is a smooth quasi-projective variety over $\mathbb{C}$ and $G$ is a finite group acting faithfully on $X$, then the Shepard-Todd theorem gives us some criterion for $X/G$ to be smooth. My ...
11
votes
1answer
361 views

Is every finite group a proper quotient of a finite primitive group?

Let $G$ be a finite group. Is there necessarily a finite primitive permutation group $P$ and a normal subgroup $N>1$ of $P$ such that $P/N \cong G$? If not, what restrictions are there on ...
11
votes
2answers
799 views

The maximum order of finite subgroups in $GL(n,Q)$

For several years people have tried to characterize finite groups of maximal order in $\mathrm{GL}(n,\mathbb{Q})$ (or $\mathrm{GL}(n,\mathbb{Z})$) and their orders. It appear in many articles a ...
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 ...
6
votes
1answer
506 views

Are there workable algebraic geometry approaches for the pentagon equation?

A pentagon equation is a system of polynomial equations of degree $3$ with several variables and integer coefficients, given by a fusion ring. A fusion ring is given by a finite set of integer ...
5
votes
1answer
572 views

Classification of $p$-groups of order $p^n$ with rank $n-1$

Hello, i've been looking for a way to classify the non-trivial $p$-groups $G$ that live in an exact sequence of the form $ 0 \rightarrow \mathbb{Z}/p\mathbb{Z} \rightarrow G \rightarrow ...
4
votes
1answer
273 views

Which finite groups can be characterized by their subgroup orders?

Given a finite group $G$, we denote by $\pi_s(G)$ the set of orders of its subgroups. Which finite groups $G$ can be characterized by the set $\pi_s(G)$, i.e. $\pi_s(H)=\pi_s(G)$ implies $H\cong G$? ...
4
votes
5answers
629 views

Product of conjugacy classes - is there an analog of Tanaka-Krein reconstruction ?

Consider a finite group G. The product of conjugacy classes can be defined in natural way just by multiplying the representatives and counting multiplicities (see e.g. MO 62088). So we get ring with ...
6
votes
2answers
259 views

Subgroup property stronger than being characteristic

In what follows, all groups are assumed to be finite. Recall that if $K \leq H \leq G$ are groups, $K$ is said to be a weakly closed subgroup of $H$ in $G$ if, for all $g \in G$, $g^{-1}Kg \leq H$ ...
5
votes
5answers
841 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
5
votes
2answers
640 views

Parabolic induction GL(n,Zp)

Let $P$ be a parabolic subgroup of $GL(n)$ with Levi decomposition $P =MN$, where $N$ is the unipotent radical. Let $\pi$ be an irreducible representation of $M(\mathbf{Z}_p)$ inflated to ...
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
1answer
306 views

Does there exist an order in a number field of deg>1 with a map to F_p for all p?

This question is motivated by a computational issue. Suppose $R$ is a product of orders in numberfields such that there is no ring homomorphism $R \to \mathbb Z$, then can one write an algorithm that ...
2
votes
0answers
329 views

A question about finite groups (a weak version of the converse of Lagrange theorem) [closed]

Let $G$ be a group of order $n$ and $d$ a positive divisor of $n$. Is it true that there exists a subgroup of $G$ with order $d$ or $n/d$? Of course this does not hold in full generality. -- In ...
6
votes
1answer
229 views

Reference for embeddings of reflection groups (related to folding ADE Coxeter graphs?

There are a couple of indirect methods, using Lie theory or Springer's general theory of regular elements in (real, complex) reflection groups, to construct natural embeddings among certain Weyl ...
3
votes
1answer
392 views

Special automorphisms of extraspecial groups

Let $G$ be an extraspecial group of order $p^{2r+1}$ (where $p$ is an odd prime), and let $V$ be a faithful representation of $G$. Consider the normal subgroup $H$ of $Aut(G)$ consisting of all ...