Questions about the branch of abstract algebra that deals with groups.

**4**

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57 views

### Examples for Bogomolov multiplier of finite group

Take $G$ to be a finite group. The projective representations of $G$ are classified by the group cohomology $H^2(G,U(1))$. We focus on the restriction map
$H^2(G,U(1)) \xrightarrow[]{\rm restriction} ...

**5**

votes

**0**answers

102 views

### Direct proof of “Nuclear implies $C_{red}^*(G) \cong C^*(G)$”

It is well-known that for a discrete group $G$ the following statements are equivalent:
$C_{red}^*(G)$ nuclear
$C_{red}^*(G) \cong C^*(G)$ canonically i.e. there exists an *-isomorphism between the ...

**-6**

votes

**0**answers

38 views

### Powerset of set and group [on hold]

Let A be some set then
(P(A),U) is a group?
where U is union operation performed on set A.
I know it is algebric structure, semi-group, monoid but not sure whether it is a group or not.
for group ...

**5**

votes

**0**answers

75 views

### Automorphism groups of cocompact Fuchsian groups as mapping class groups

Let $\Gamma$ be a cocompact Fuchsian group. So it has presentation $$\langle x_1,y_1, \dots, x_g,y_g,z_1, \ldots, z_r \mid [x_1,y_1] \cdots [x_g,y_g]z_1 \cdots z_r=1, \ z_i^{m_i}=1 \rangle$$
for some $...

**6**

votes

**1**answer

203 views

### A combinatorial property of uncountable groups, II

Problem 1. Is it true that each uncountable group $G$ contains two subsets $A,B\subset G$ such that
1) for any $x,y\in G$ the intersection $xA\cap yB$ is finite and
2) for any function $\...

**2**

votes

**0**answers

44 views

### One-parameter group of nonvanishing vector field

Let $M$ be a smooth manifold( if necessary one can assume it is closed), $V$ be a non-vanishing vector field of $TM$.
Q: Under what condition, we can say that $\overline{\exp(tV)}$, i.e. the closure ...

**6**

votes

**1**answer

172 views

### A combinatorial property of uncountable groups

Let $A,B$ be two uncountable sets in a group $G$ such that for any elements $x,y\in G$ the intersection $xA\cap yB$ is finite. Let $\Phi:G\to 2^G$ be a function assigning to each element $x\in G$ some ...

**5**

votes

**0**answers

48 views

### Structure of invariant lattices and reductions of group representations with $\text{dim}>2$

Let $G$ be a finite group, $K$ be a $p$-adic field with an uniformizer $\pi$ and residue field $k \cong \Bbb F_q$, and $V$ be an irreducible representation of $G$ over $K$. Consider $X_{V}^G$ the set ...

**1**

vote

**0**answers

70 views

### What is an upper limit of relative size of conjugacy class of the transitive finite group?

What is
$$ \limsup_{n\to\infty} \sup_{\deg(G)=n} \left( \frac{\max_{x\in G} \left|\operatorname{conj}(x)\right|}{\operatorname{ord}(G)}\right),$$
$G$ transitive permutation group?
And what are the ...

**1**

vote

**0**answers

54 views

### Weyl Group action on the complement of the Tits Cone in a Kac-Moody algebra

Given a Kac-Moody algebra $\mathfrak h$ and its Weyl group $W$, the action of $W$ on the Tits cone $X$ is well understood. Decompose $\mathfrak h$ into $X\cup -X\cup L$. Then the action of $W$ on $-...

**12**

votes

**1**answer

211 views

### Equivalence of surjections from a surface group to a free group

Let $g \geq 2$. Let $S = \langle a_1,b_2,...,a_g,b_g | [a_1,b_1] \cdots [a_g,b_g] \rangle$ be the fundamental group of a genus $g$ surface and let $F_g$ be a free group with $g$ generators. Given ...

**17**

votes

**0**answers

304 views

### Does $\mathrm{SL}_{n}(\mathbb{Z}/p^{2})$ have the same number of conjugacy classes as $\mathrm{SL}_{n}(\mathbb{F}_{p}[t]/t^{2})$?

Let $p$ be a prime; $\mathbb{F}_{p}$ is the field with $p$ elements
and $\mathbb{F}_{p}[t]$ the ring of polynomials in $t$ over $\mathbb{F}_{p}$.
Does $\mathrm{SL}_{n}(\mathbb{Z}/p^{2})$ have the ...

**3**

votes

**2**answers

181 views

### Casimir operator of a given Lie algebra and relation with its matrix representation

I'm following Gilmore's recipe to compute the abstract Casimir operator of a given algebra (in this example, I refer to algebra su(2)). This recipe bring up a matrix representation of the algebra and ...

**1**

vote

**0**answers

54 views

### Walk in the graph induced by a group action

Suppose that graph $G$ is induced by a group $⟨α_1,...,α_r⟩$ acting on a large finite set $X$ for small $r$. To be precise, we have the vertex set $V(G):=X$,
and $x_1x_2\in E(G)$ whenever for some $\...

**17**

votes

**1**answer

366 views

### Probability of satisfying a word in a compact group

This question is inspired by Probability of commutation in a compact group, which asked whether $P(xyx^{-1}y^{-1} = 1)$ could take values strictly between $0$ and $1$ on a compact connected group. ...

**3**

votes

**0**answers

69 views

### character table of $2D_{8}(2)$

In my new research study I need the character tables of a simple group $^2D_8(2)$ and for that I was trying to find a reference and in this regard I came across a reference "Character degrees and ...

**3**

votes

**1**answer

102 views

### On decomposition of finite Abelian groups

It is easy to see that for any finite Abelian group $G$ and any numbers $a,b$ with $|G|=ab$ there exist a subgroup $A\subset G$ and a subset $B\subset G$ such that $|A|=a$, $|B|=b$ and $G=A+B$, where $...

**13**

votes

**3**answers

706 views

### Probability of commutation in a compact group

It is well known that if $G$ is a finite group, then the probability that two elements commutte is either $1$ (if $G$ is abelian) or less than or equal to $\frac58$.
If instead $K$ is a compact group,...

**4**

votes

**0**answers

127 views

### Uniqueness of the boundary of a hierarchically hyperbolic group

Hierarchically hyperbolic groups and spaces (HHG and HHS for short) were defined by Behrstock, Hagen and Sisto (see here and here). Examples include mapping class groups, Right angled Artin groups, ...

**7**

votes

**1**answer

135 views

### How should I think about the module of coinvariants of a $G$-module?

Let $G$ be a group, $M$ a $G$-module, then the group of coinvariants is the module $M_G := M/I_GM$, where $I_G$ is the kernel of the augmentation map $\epsilon : \mathbb{Z}G\rightarrow \mathbb{Z}$.
...

**3**

votes

**1**answer

103 views

### Is it possible to classify extensions $G$ of an abelian $A$ by an abelian $N$ such that the map $G\rightarrow A$ is abelianization?

Suppose we have a given action $\varphi : A\rightarrow\text{Aut}(N)$ with $A,N$ abelian groups. Is it possible describe the isomorphism classes of extensions $G$ of $A$ by $N$ realizing $\varphi$ such ...

**5**

votes

**0**answers

140 views

### Are these element in a group algebra of a torsion-free group zero divisors?

Let $G$ be an arbitrary torsion-free group. For $x,y\in G$, which of these elements can be decided immediately not to be zero divisors in $\mathbb ZG$ (or in $\mathbb CG$)?
$$1+x+y,\quad 4+x+x^{-1}+y+...

**14**

votes

**0**answers

172 views

### Young's natural representation of the symmetric group

The literature on the representation theory of the symmetric group contains some terminology that I find puzzling, and I am wondering if someone here knows the full story.
One of the standard ways to ...

**4**

votes

**1**answer

173 views

### Determining the conjugacy classes of a wreath product $G \wr S_n$

If $G$ is a finite group and its conjugacy classes are known, can the conjugacy classes of the wreath product $G \wr S_n \cong G^n \rtimes S_n$ be determined?

**3**

votes

**0**answers

160 views

### Representations of GL(n,2) over a field of characteristic 2

I would appreciate very much if you can point to me some references on the following:
1) Representations of the linear group $GL(n,2)$ over $F_2$.
2) Representations of $GL(n,2)$ over an algebraic ...

**5**

votes

**1**answer

112 views

### Is there a subgroup of dual depth 3?

This post is motivated by an exchange with Zhengwei Liu. It is more than the dual version of this post, because we consider any subgroup (instead of just maximal), and even more at the end...
Let's ...

**4**

votes

**1**answer

193 views

### Hereditarily indecomposable groups

Question. Is it true that each uncountable group $G$ contains an uncountable subgroup $A$ and an infinite subgroup $B$ such that $A\cap B=\{1\}$? What will be the answer if we additionally require ...

**0**

votes

**0**answers

32 views

### Reference request: Limit independent of Følner sequence

Consider Theorem 1 in http://prac.im.pwr.edu.pl/~downar/conferences/slides/banska-bystrica.pdf . It roughly says that for a discrete group $G$, the limit
$$ \lim f(\Lambda_n) / \lvert \Lambda_n \rvert$...

**1**

vote

**0**answers

232 views

### Some questions on linear algebraic groups and their eigenvalues

Let $G$ be a connected linear algebraic group (a torus, for example) of finite type over a field $K$. I have several, I suppose rather base, questions concerning the theory of such groups.
Suppose ...

**9**

votes

**2**answers

420 views

### About normal minimal subgroups not in the Frattini

In Neukirch--Schmidt--Wingberg, "Cohomology of Number Fields", Second edition, page 624, Exercise 2, it is stated the following fact.
$\textbf{Claim}$: If $N$ is a normal subgroup, minimal among ...

**19**

votes

**3**answers

766 views

### Does the hypergraph of subgroups determine a group?

A hypergraph is a pair $H=(V,E)$ where $V\neq \emptyset$ is a set and $E\subseteq{\cal P}(V)$ is a collection of subsets of $V$. We say two hypergraphs $H_i=(V_i, E_i)$ for $i=1,2$ are isomorphic if ...

**9**

votes

**0**answers

167 views

### Continuous cohomology of a profinite group is not a delta functor

Let $G$ be a profinite group, then there is a general notion of continuous cohomology groups $H^n_{\text{cont}}(G, M)$ for any topological $G$-module $M$ (I require topological $G$-modules to be ...

**6**

votes

**3**answers

287 views

### Is there a maximal subgroup of depth 3?

Let's first define what we mean by depth of a subgroup.
Let $G$ be a finite group and $H$ a subgroup. Let $(V_i)_{i \in I}$ and $(W_j)_{j \in J}$ be the irreducible complex representations of $G$ ...

**5**

votes

**0**answers

123 views

### The preimage of bounded real intervals under homomorphisms on hyperbolic groups

Let $G$ be a hyperbolic group with a fixed (finite, symmetric) generating set and suppose that $\varphi : G \to \mathbb{R}$ is a group homomorphism. Write $W_n = \{ g \in G: |g|=n\}$, where $|g|$ ...

**4**

votes

**0**answers

117 views

### Appearances of $\mathbb{Q}/\mathbb{Z}$ in Pontryagin duality for profinite groups

(This is a somewhat lazy question which came up as I'm reading about Pontryagin duality for the first time)
For a locally compact abelian topological group $G$, its Pontryagin dual is the group of ...

**6**

votes

**4**answers

485 views

### What is a geodesic in Outer space?

The Culler-Vogtmann Outer space $\text{CV}_n$ is an analogue of Teichmuller space for the group $\text{Out}(F_n)$.
Is there any notion of a geodesic path in $\text{CV}_n$? Are there different ...

**5**

votes

**0**answers

67 views

### A Spatial-Orientation Counting Problem

Suppose I have 36 black blocks of dimensions 1x2x3. I can stack them 2 across, 3 deep and 6 high to make a nice looking cube of dimensions 6x6x6. I then proceed to paint the surface of this cube red. ...

**2**

votes

**0**answers

81 views

### A survey on locally finite groups

I have some idea that might be useful for locally finite groups. However, I realised that except for the definition I know almost nothing about locally finite groups. Is there any place I can find a ...

**5**

votes

**0**answers

164 views

### Projecting GxG onto subspace with tied irreducible representations

Suppose I have a finite group $G$. With this group, I can associate an ortho-normal Hilbert space spanned by elements of the group $$\mathcal{H} = \{|g\rangle: g \in G \}$$.
I could alternatively ...

**12**

votes

**0**answers

121 views

### Open questions about automorphism tower theorem

Joel Hamkins has left four open questions about automorphism tower theorem in his wonderful paper Every group has a terminating transfinite automorphism tower.
In fact, the four questions are "For ...

**7**

votes

**0**answers

164 views

### Number of elementary abelian subgroups of extraspecial $2$-groups

Let $G$ be an extraspecial $2$-group, i.e. $Z(G)=G'=\Phi(G)$ has order $2$. Then $|G|=2^{2n+1}$ for some $n\geq 1$, $G\cong D_8^{*n}$ or $G\cong Q_8*D_8^{*(n-1)}$, and $G$ has one of the following ...

**3**

votes

**0**answers

76 views

### Enlarging a subdegree-finite “almost transitive” permutation group to a transitive one? (follow-up)

Consider a permutation group $G$ acting on an infinite set $X$. Assume $G$ has finitely many orbits, and every point stabiliser $G_x$ has finite orbits.
Can we always find a permutation $\tau\in\...

**4**

votes

**1**answer

282 views

### What is the asymptotic growth of $\sum_{k=1}^n 2^{\omega_k}$?

Question: Let $\omega_k$ be the number of distinct prime divisors of k.
What is the asymptotic growth of $C_n := \sum_{k=1}^n 2^{\omega_k}$?
Thank you for considering this elementary question. ...

**9**

votes

**1**answer

272 views

### Are finite presentations of arithmetic groups computable?

In this famous paper by Borel and Harish-Chandra, Arithmetic Subgroups of Algebraic Groups, it is proved that, in characterisitic zero, arithmetic groups are finitely presented. I have an extremely ...

**1**

vote

**1**answer

82 views

### Study of the subgroups of which a non-linear monomial character is induced from

Let G be a monomial group, and let H_1,...,H_r be the subgroups of G where there exists a linear character that induces to an irreducible character of G. How much is known about these subgroups? For ...

**2**

votes

**1**answer

81 views

### Enlarging a subdegree-finite “almost transitive” permutation group to a transitive one?

Consider a permutation group $G$ acting on an infinite set $X$. Assume $G$ has finitely many orbits, and every point stabiliser $G_x$ has finite orbits. Now consider a permutation $\tau\in\...

**5**

votes

**2**answers

190 views

### Is there a size 2 generating set of the signed symmetric group $B_n$?

The signed symmetric group $B_n$ is a permutation group where the underlying set is $B_n=\{\sigma \in S_{A_n}| \forall x \in A_n, \sigma(-x)= -\sigma(x)\}$ with $A_n=\{-n,-(n-1),-(n-2),\cdots,-1,1,\...

**4**

votes

**1**answer

166 views

### Database subgroups of free group

Is there some database that contains "all" low-index normal subgroups of the free group on two generators?
Extension: does there exist such a GAP-database?
Thank you!

**2**

votes

**1**answer

238 views

### About the growth rate of a group

Let $G$ be a f.g. group and $d$ be a word metric w.r.t. a symmetric generating set. For $g\in G$, define $|g|:=d(g,e)$, where $e$ is the group identity. For $k\in\mathbb N$, put
$$n_k:=\#\{g\in G: |g|...

**12**

votes

**0**answers

257 views

### Escaping from a centralizer

Let $G = Sym(n)$, $n$ even. Let $H<G$ be the stabilizer of the partition $\{\{1,2\},\{3,4\},\dotsc,\{n-1,n\}\}$, or, what is the same, the centralizer of $(1\;2) \dotsc (n-1\; n)$.
By Stirling's ...