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

**2**

votes

**0**answers

58 views

### Actions on spaces with measured walls

In geometric group theory, the question of whether or not a group acts nicely on a CAT(0) cube complex, or equivalently on a median graph, is of interest. The same question for actions on spaces with ...

**5**

votes

**1**answer

170 views

### Do one-relator groups satisfy Haagerup property?

The question is in the title:
Do one-relator groups satisfy Haagerup property?
I think the answer is known at least in some specific cases, but is the problem completely solved?

**9**

votes

**1**answer

238 views

### Is the free product $\mathbb{Z}*\mathbb{Z}/n\mathbb{Z}$ linear over $\mathbb{Z}$?

Let $H:=\mathbb{Z}*\mathbb{Z}/n\mathbb{Z}=\langle p,q| q^n=1\rangle.$ I want to know if $H$ is a ($\mathbb{Z}$)linear group that is to say is there an injective homomorphism $f: H\to GL_m(\mathbb{Z})$ ...

**34**

votes

**0**answers

952 views

### $A$ is isomorphic to $A \oplus \mathbb{Z}^2$, but not to $A \oplus \mathbb{Z}$

Are there abelian groups $A$ with $A \cong A \oplus \mathbb{Z}^2$ and $A \not\cong A \oplus \mathbb{Z}$?

**16**

votes

**0**answers

512 views

### How boundedly generated is $SL_3(\mathbb{Z})$?

The group $G = \mathrm{SL}_3(\mathbb{Z})$ is known to be boundedly generated, that is, there exists some $m \in \mathbb{N}$, and $g_1, \dots, g_m \in G$ such that we have the following equality of ...

**4**

votes

**0**answers

72 views

### Methods for showing homology of a subgroup survives to the larger group

Suppose we have an inclusion of groups $G_1<G_2$. I am curious about what methods there are out there for analyzing the map $H_k(G_1;\mathbb Q)\to H_k(G_2;\mathbb Q)$. In particular, what are tools ...

**4**

votes

**0**answers

79 views

### Smooth admissible representations, Hom, tensor and extension of scalars

(Remark: This has previously been posted on math.stackexchange, but I believe it might be suitable for this site as well. ...

**4**

votes

**2**answers

354 views

### What does it mean to take the diagonal of the group $SU(2) \times SU(2) $?

I am reading Witten's paper on topological field theories, in specific the topological twist in page 359. In order to perform the twist he takes the diagonal subgroup of $K = SU(2)_{\text{Right}} ...

**7**

votes

**2**answers

310 views

### Theorems of the Galois groups of quintics appears not to work for the ${F}_{20}$ group determination

I am computing the Galois groups of quintics using the theorems from Ryan Kavanagh paper "On Irreducible Rational Quintics" using the decic resolvent ${P}_{10} \left({x}\right) = \prod\limits_{1 \le i ...

**4**

votes

**2**answers

310 views

### Explicit description of the principal block of the symmetric group

Let $k$ be a field of prime characteristic $p$ and $\Sigma_n$ be the symmetric group.
If I have a concrete $k[\Sigma_n]$-module $M,$ how to compute the direct summand corresponding to the principal ...

**3**

votes

**1**answer

158 views

### Is there a bound on the rank of finite index subgroup of SL_3(Z)?

Is there an $N \in \mathbb{N}$ such that every finite index subgroup of $\mathrm{SL}_3(\mathbb{Z})$ has a generating set of size $N$?

**4**

votes

**1**answer

94 views

### Is being Noetherian a quasi-isometric invariance for f.g. groups?

Recall that a group $G$ satisfies max (or is said to be Noetherian) if all its proper subgroups are finitely generated. Similarly $G$ satiesfies max-n if all its normal subgroups are normal closures ...

**5**

votes

**1**answer

169 views

### Abelianization of characteristic quotients of $F_2$

Let $F_2$ be the free group on two generators.
Let $U\le F_2$ be a characteristic subgroup of finite index, and let $f : F_2\rightarrow\mathbb{Z}^2$ be the abelianization map.
It's easy to check ...

**9**

votes

**2**answers

224 views

### Linear occurrences of finite simple groups

Let $S$ be a finite simple group. All representations below are over the complex numbers.
Let
$d_0(S)$ be the smallest dimension of a faithful representation of $S$,
$d_1(S)$ be the smallest ...

**5**

votes

**1**answer

347 views

### Can a group be a union of finitely many subgroups of infinite index?

Is there a group $G$ and subgroups $H_1, \dots, H_n \leq G$ for some $n \in \mathbb{N}$, such that $[G : H_i] = \infty$ for each $1 \leq i \leq n$, and $$G = \bigcup_{i=1}^n H_i \ \ ?$$

**3**

votes

**1**answer

102 views

### Character degrees of extensions of 2^B_2(q^2)

The outer automorphism group of the Suzuki simple group
${}^2B_2 (2^{2m+1})$, $m \geq 1$ is cyclic of order $2m+1$ and
is generated by a field automorphism $\varphi$ of order $2m+1$.
For any ...

**2**

votes

**0**answers

125 views

### A question of braid words

Let $(W,S)$ be a Coxeter group, let $B(W,S)$ be the corresponding braid or Artin-Tits group. Set $S=\{s_1,\dots, s_n\}$ and denote by $\bf{S}=\{\sigma_1,\dots, \sigma_n\}$ the corresponding generators ...

**3**

votes

**0**answers

388 views

### Some counter examples in group theory

In this question, which we flag it as a community wiki question, we search for a big list of groups $G$ which can not be isomorphic to a structure mentioned in $i.$ for some $i \in ...

**5**

votes

**2**answers

322 views

### Is there a nonabelian free group inside a group of positive rank gradient?

Let $G$ be a finitely generated residually finite group with positive
rank gradient, and let $F_2$ be the free group on $2$ elements. Must
there be an embedding $i \colon F_2 \to G$ ?
A group ...

**9**

votes

**2**answers

108 views

### Generate polyhedra by collapsing vertices of a polyhedron

I am looking for basic information about the following idea:
(I) Consider a square. By collapsing two adjacent vertices, we obtain a triangle.
(II) Consider a three-dimensional cube. By collapsing a ...

**8**

votes

**2**answers

293 views

### Fell topology vs. convergence of matrix coefficients

My question is partially inspired by the following discussion:
Topology on the Unitary Dual
Let me remind/explain how the Fell topology is defined (at least I recall the definition which I saw): let ...

**6**

votes

**2**answers

203 views

### Zero divisors with support of size 3 in group algebras of finite groups

Are there a finite group $G$ and a field $\mathbb{F}$ such that $\gcd(3,|G|)=1$ and the group algebra $\mathbb{F}[G]$ contains a zero divisor whose support is of size $3$?
Recall that the support of ...

**4**

votes

**0**answers

119 views

### largest subgroup of $Out(\hat{F_2})$ which preserves the Nielsen invariant

Let $x,y$ be generators for the free group $F_2$. It's known that $Aut(F_2)$, and hence $Out(F_2)$ preserves the conjugacy class of the subgroup $\langle[x,y]\rangle$ generated by $[x,y]$ (This ...

**4**

votes

**0**answers

74 views

### Roller's problem on median groups

At the end of his dissertation Poc Sets, Median Algebras and Group Actions, Martin Roller asks
A group $G$ is called median if it acts freely and transitively on a median algebra. This is ...

**1**

vote

**1**answer

259 views

### Does $\mathbb Z \times \mathbb Z$ mod the obvious $\mathbb Z$ action have more structure than just a set?

$\mathbb Z$ acts on the lattice $\mathbb Z \times \mathbb Z$ by adding an element to itself n times.
I am studying some function arising from symplectic geometry which happens in my case to be ...

**1**

vote

**2**answers

141 views

### Automorphism group of the affine groups AGL(n,q), ASL(n,q)

I have a question. The automorphism group of the linear groups $GL(n,q)$, the group of linear transformations of $V = \mathbb{F}_q^n$, and $SL(n,q)$, the subgroup of $GL(n,q)$ consisting of elements ...

**5**

votes

**1**answer

178 views

### Generalized identities of (soluble) groups

Let $G$ be a group. Let us say that $G$ satisfies a generalized identity of degree $n$ if there exist $a_1,a_2,\dots a_n \in G$ such that
$$x^{a_1}x^{a_2}\dots x^{a_n}=1,$$
for all $x\in G$.
...

**5**

votes

**0**answers

131 views

### Indecomposable representations of a wreath product

If $G$ is a finite group, we know the irreducible representations of $G ≀ S_n$ (over $\mathbb Q$) are classified by partitions of $n$ 'decorated' by an irrep of $G$.
I'm wondering to what extent the ...

**8**

votes

**2**answers

533 views

### Does the Galois group of a Pisot polynomial contain the alternating group?

Let $n \in \mathbb{N}$, and let $p(X) \in \mathbb{Z}[X]$ be a monic polynomial of degree $n$. Suppose that exactly one complex root of $p$ is of modulus $> 1$, and that the remaining $n-1$ roots of ...

**2**

votes

**2**answers

376 views

### An approximate version of $g^2 = e$ for all $g \in G$, implies $G$ is Abelian [closed]

I am trying to understand the concept of approximate group.
So I took a group theory exercise from a physics class at Caltech. The question basically states:
Suppose that for any element $g ...

**12**

votes

**1**answer

287 views

### Applications of Lubotzky's linearity theorem?

Lubotzky's theorem is a necessary and sufficient set of conditions for a finitely generated discrete group to be linear, i.e. isomorphic to a subgroup of $GL_n(K)$, where $K$ is a field of ...

**2**

votes

**1**answer

207 views

### Any representation is a sub representation of direct sum of regular representation

I need a reference for the following statement:
Let G be a linear algebraic group over algebraically closed field k. Let V be a finite dimensional G-module. The V is sub representation of k[G]^n for ...

**4**

votes

**0**answers

107 views

### do there exist finite simple characteristic quotients of the free group of rank 2?

Let $F_2$ be the free group of rank 2. Let $Aut^+(F_2)$ be the subgroup of $Aut(F_2)$ consisting of automorphisms of determinant 1 under abelianization.
Do there exist maximal normal finite index ...

**8**

votes

**1**answer

333 views

### Three involutions on the set of 6-box Young diagrams

The set of $n$-box Young diagrams classifies both conjugacy classes in $S_n$ and equivalence classes of irreducible representations of $S_n$. There is an outer automorphism of $S_6$, of order 2. ...

**6**

votes

**1**answer

337 views

### Some questions about the Malcev completion

Let $G$ be an abstract group. The Malcev completion $\widehat{G}$ of $G$ (over $\mathbb{Q}$) is the set of group-like elements in the complete Hopf algebra $\widehat{\mathbb{Q}[G]} = \lim_n ...

**3**

votes

**1**answer

276 views

### Finitness of the Burnside Group

This is something I discussed with Andrezj Zuk, but we didn't arrive to any conclusions. Let $B(d,n)$ be the Burnside group on $d$ generators of exponent $n$. Is there an algorithm to determined ...

**3**

votes

**1**answer

113 views

### Transitive actions of $Aut(F_2)$ on surjections from $F_2\twoheadrightarrow G$

Here $F_2$ is the free group on two generators $x,y$. I'm interested in examples of finite groups $G$ such that $Aut(F_2)$ acts transitively on the set of surjections $F_2\rightarrow G$. (In ...

**5**

votes

**1**answer

393 views

### Can the Kan-Thurston theorem be turned into some kind of equivalence between groups and spaces?

I not really familiar with these subjects. I read this question and I was really surprised by the answer. My question is probably vague (so please do bear with me).
The cited question/answer ...

**2**

votes

**2**answers

133 views

### Are the Baumslag-Solitar groups BS(n,n) and BS(n,-n) automata groups?

In this article of Bartholdi and Sunik http://arxiv.org/abs/math/0603032, they say that BS(n,n) and BS(n,-n) are automata groups because they are virtually $F_{|n|}\rtimes\mathbb{Z}$ (where $F_{|n|}$ ...

**28**

votes

**2**answers

1k views

### How much of the ATLAS of finite groups is independently checked and/or computer verified?

In a recent talk Serre made some comments about proofs that rely on the classification of finite simple groups (CFSG) and on the ATLAS of Finite Groups. Namely, he said that a proof that relied on the ...

**2**

votes

**2**answers

151 views

### Detecting HNN-Extension and free products with amalgamation

This question is partly connected with the following Connection between Stalling's end theorem and Seifert-van Kampen Theorem.
By Stalling's Theorem a group with more than one end splits over a ...

**2**

votes

**1**answer

182 views

### How bad can an infinite linear torsion group be?

Following this question I wonder about the following. Examples of infinite torsion groups which are linear in zero characteristic are infinite groups of roots of unit.
Are there other examples ...

**2**

votes

**1**answer

188 views

### Finite sequences which happen to be the sequence of orders of elements of a simple group

Let $n\in \Bbb N$. Any finite group $G$ with $|G|=n$ has a solution for the Diophantine equation
$$\sum_{d|n}x_d\phi(d)=n~~~~~~~~~~~~~~~~~~~~~~~(1)$$
where $\phi$ is the Euler's totient function, $d$ ...

**3**

votes

**1**answer

186 views

### variance of the number of fixed points for a permutation group

It is reasonably well-known that the variance of the number of fixed points for $S_n$ equals $1.$ Now, what about other transitive permutation groups on $\{1, \dotsc, n\}?$ Presumably much is known. I ...

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votes

**1**answer

78 views

### On Sylow subgroup of a finite group [closed]

Let $p\mid n$, then by $n_p$ we mean the $p$-part of $n$, i.e. $n_p = p^k$ if $p^k\mid n$ but $p^{k+1}\nmid n$. Let $G$ be a finite group, $M\leq G$ and $P\in Syl_p(G)$. Is It true that $|M\cap ...

**3**

votes

**1**answer

112 views

### Classification of finite abelian hypergroups and table algebras

Update: Originally, I formulated this question for finite abelian hypergroups, but in a discussion with Geoff Robinson below I realized that the abelian hypergroups defined below are equivalent to ...

**2**

votes

**0**answers

97 views

### Obtaining a quasi-isometry of the 'boundary'

It is well-known that a quasi-isometry induces a homeomorphism on the space of ends of say a locally finite graph for simplicity. Clearly the converse is not true. In other words the concept of ends ...

**3**

votes

**1**answer

137 views

### Connection between Stalling's end theorem and Seifert-van Kampen Theorem

Stalling`s end Theorem (a group has more than one end iff it splits over a finite subgroup) and the Seifert-van Kampen Theorem (the fundamental group of a 'decomposable' space is a free amalgamated ...

**3**

votes

**1**answer

248 views

### Infinite groups of finite exponent inside of SL(2,C)

Fix an integer $n>0$. Are there infinite subgroups of $SL_2(\mathbb{C})$ such that every element is $n$-torsion?

**3**

votes

**0**answers

263 views

### Maximizing sum of matrices

Over the last few months, I've been trying to find the solution to a research-related problem I'm having. However, my research is not in mathematics, and my progress toward reaching a solution has ...