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

**4**

votes

**0**answers

141 views

### Is there a nontrivial profinite word which is trivial in any group with at most d generators?

Let $F$ be a free profinite group of rank $\aleph_0$, and let $d \in \mathbb{N}$.
Let $N_d \lhd_c F$ be the intersection of all open normal subgroups $L \lhd_o F$ for which $F/L$ can be generated by ...

**8**

votes

**2**answers

299 views

### Dehn algorithm and normal forms in small cancellation groups

I found this statement in B. Cavallo, D. Kahrobaei's paper arXiv:1311.7117
Secret Sharing using Non-Commutative Groups and the Shortlex Order, page 7.
"C′(1/6) continue to be an ideal platform for ...

**1**

vote

**0**answers

45 views

### Exhausting a free pro-p group

Recall that for a profinite group $G$ we define the subgroup rank to be $$\sup \{d(H): H \leq_c G\}$$ where $d(H)$ stands for the minimal cardinality of a set of topological generators of $H$.
Let ...

**3**

votes

**1**answer

203 views

### Base for symmetric group

Given symmetric group $S_n$, is it possible to find $k=\lceil\log_2S_n\rceil=\lceil\log_2n!\rceil$ members $\{\alpha_i\}_{i=1}^{k}$ in $S_n$ such that every member of $S_n$ can be written as ...

**0**

votes

**1**answer

92 views

### Example of a polycyclic group which is not of polynomial growth? [closed]

The title already says everything: What is an example of a polycyclic group $G$ which is not of polynomial growth (equivalently, by Gromov's theorem, which is not virtually nilpotent)?

**1**

vote

**0**answers

82 views

### Conjugacy classes in lie type group

I have two questions. Thanks for any comments.
Suppose $S$ is a simple group of Lie type in characteristic p. Also suppose that $G=Aut(S)$.
1) Is there any reference for conjugacy class of element ...

**6**

votes

**1**answer

207 views

### Action of the homotopy braid groups on reduced free groups

Firstly some definitions:
$B_n$ is the braid group with $n$ strands.
$\widetilde{B_n}$ is "homotopy braid group", which is a factor group of $B_n$ by adding the relation that $A_{j,k}$ ...

**1**

vote

**0**answers

88 views

### Selecting dense diagonals in $\Bbb T^2$

Let $p$ be a prime number and let $G=\bigcup_{n\in \Bbb N}\{\exp(k\frac{2\pi i}{p^n})\mid k\in \Bbb Z\}$ be a Prüfer group. For homomorphisms $f,g:G\to G$ let $H_{f,g}=\{(f(x),g(x))\mid x\in G\}$. ...

**0**

votes

**0**answers

68 views

### faithful action of Hecke algebra

Let $G$ be a connected reductive group split over a number field $F$, $\mathbb{A}$ the adeles.
Let $v$ be a finite place and $\mathcal{H}_{v}$ the spherical hecke algebra at palce $v$.
...

**0**

votes

**0**answers

96 views

### Automorphism of simple lie type groups

I will be so thankful for any comment or answer.
Suppose $S$ is a simple Lie type group of characteristic $p$ and $S\subseteq G \subseteq Aut(S)$ and $G_0$ is a subgroup of $G$ generated by all inner ...

**4**

votes

**0**answers

121 views

### Is it provable in $\mathsf{ZF}$ that there is a group structure on any set $X$? [duplicate]

Given a set $X$ is it provable in $\mathsf{ZF}$ that there is a binary operation $\ast: X\times X\to X$ such that $(X,\ast)$ is a group?

**4**

votes

**1**answer

384 views

### Generalization of a property of $A_n; n\geq 5$

Let $H$ and $K$ be two proper non-trivial subgroups of the
alternating group $A_n; n\geq 5$.
Then there exists a maximal subgroup $M$ of $A_n$
such that $H\not\leq M$ and $K\not\leq M$.
To see this
...

**1**

vote

**1**answer

139 views

### abelian p- subgroups of E_6(q)

Is there any result about maximal abelian p-subgroups of the exceptional group E_6(q), where q=p^a is prime power?

**0**

votes

**0**answers

81 views

### Adelic integral factorization

In order to calculate Tamagawa numbers, I need to justify that for a nice (say Schwartz-Bruhat) function, the following identity holds :
$$\int_{\mathbf{A}^2} f(x)dx = \int_{SL_2(\mathbf{A})/SL_2(K)} ...

**1**

vote

**1**answer

162 views

### Can we say that $A$ is a complement for a group $G$?

Let $A$ be a frobenius complement for a group $G$ i.e. $A$ act on $G$ by automorphism s.t. $C_A(g)=e$ for all nonidentity $g$.
Now, Action of $A$ can be linearly extended so that $A$ act on $F[G]$. ...

**1**

vote

**0**answers

65 views

### Periodic Growth behaviours of Cayley graphs

This question is related to
On the size of balls in Cayley graphs
and
Folner sequences of amenable groups of exponential growth
Given a Cayley graph of a group $G$ with finite generating set ...

**5**

votes

**1**answer

306 views

### Groups with a unique composition series

Which finite groups $G$ have a unique composition series? I don't mean in the sense of the Jordan-Holder theorem, but rather actually unique.
Some examples are the cyclic groups $C_{p^n}$ and the ...

**4**

votes

**2**answers

257 views

### Powers of finite simple groups

I have heard about the following result: for each finite simple non-abelian group $S$ and each natural number $r\ge 2$ there exists a number $n=n(r,S)$ such that the power $S^n$ is $r$-generator but ...

**1**

vote

**0**answers

128 views

### Finding an overgroup or a subgroup in PGL

Let k be a nonperfect field of characteristic $2$. Let $a\in k\backslash k^2$, $G=PGL_4(k)$ and and $$H= \left\{ \small\left[\begin{array}{cccc}
x ...

**4**

votes

**0**answers

178 views

### Unitary representations of Tarski Monsters and other beasts

Did people study the unitary representations of Tarsky Monsters, for example the ones constructed by Ol'shanskii? Are there any exotic representations, ie. except the ones related to the left regular, ...

**2**

votes

**1**answer

135 views

### (Alternative) Presentation for the pure braid group of the sphere

First I need some notation (it's all standard I think). For a manifold $M$, let $F_nM = F_{0,n}M$ be the space of $n$-tuples of distinct points on $M$ ; let $B_nM = B_{0,n}M = F_nM / \Sigma_n$. When ...

**1**

vote

**0**answers

164 views

### Rational conjugation of elements of a finite group

Let $G$ be a finite group. Two elements $x$ and $y$ of $G$ are said to be rationally conjugate, written $x \sim_{r} y$, if and only if $\langle x\rangle$ and $\langle y\rangle$ are conjugate subgroups ...

**3**

votes

**2**answers

176 views

### Cohomology of SL(2,R) with coefficients given by linear action

Let $SL(2,{\mathbb R})$ act on ${\mathbb R}^2$ by matrix multiplication.
What is known about group cohomology $H^*(SL(2,{\mathbb R}),{\mathbb R}^2)$?
And about $$H^*(\Gamma,{\mathbb R}^2)$$ for a ...

**8**

votes

**1**answer

203 views

### amenable + without $BS(m,n)$+finite $K(G,1)$implies virtually cyclic?

I heard from someone that the following problem is an open question.
(Open Problem 1)For a countable discrete group $G$, suppose it does not contain any Baumslag-Solitar subgroups $BS(m,n):=\langle ...

**1**

vote

**1**answer

300 views

### Word metrics and finite index subgroups

Suppose that we are given some finitely generated group $ G $ and some finite index subgroup of it $ H $. Given a finite generating symmetric generating set $ S \subset G $, we can define the word ...

**5**

votes

**3**answers

169 views

### bounding from below the cardinality of a set of generators of the $n$-fold cartesian product group of a finite group

Let $G$ be a non-trivial finite group. Let $n\in\mathbf{Z}_{\geq 1}$ and let $G^n$ be the $n$-fold cartesian group product of $G$. Let $S\subseteq G^n$ be a generating set of $G^n$.
Q: Is $|S|\geq ...

**6**

votes

**1**answer

218 views

### Are profinite groups of cardinality $|\mathbb{R}|$ determined by their finite quotients?

Question: Let $G,H$ be profinite groups of cardinality $|\mathbb{R}|$, with the same finite quotients (here I only consider quotients by normal, open subgroups). Then are $G$ and $H$ isomorphic?
...

**7**

votes

**2**answers

579 views

### Every free abelian group is slender, why?

Wikipedia states that every free abelian group is slender. Where can I find a proof?
If this is not trivial, then I will also need a reference to use in my paper.

**2**

votes

**1**answer

144 views

### Every homomorphism from the Baer–Specker group into a slender group factors through ${\bf Z}^n$, why?

Wikipedia states that every homomorphism from the Baer–Specker group ${\bf Z}^{\bf N}$ into a slender group factors through ${\bf Z}^n$ for some natural number $n$. Where can I find a proof?
If this ...

**1**

vote

**0**answers

66 views

### Lower periodic subsets of groups and semigroups

Suppose that $A$ and $B$ are subsets of a group or semigroup. We call $A$ left
upper [resp. lower] $B$-periodic if $BA\subseteq A$
[resp. $A\subseteq BA$]. If $A$ is both left upper and
lower ...

**1**

vote

**1**answer

81 views

### A condition for Hypercentral Groups to be Abelian

I'm reading an article I wrote my doctoral supervisor. In this article he states that if $G$ be a hypercentral group and suppose that $G$ is generated by (a finite number of) Prufer subgroups. Then ...

**4**

votes

**1**answer

180 views

### Are homotopy braid groups residually nilpotent?

A group is called residually nilpotent if given any non-identity element, there is a normal subgroup not containing that element, such that the quotient group is nilpotent. It is known that pure braid ...

**8**

votes

**0**answers

230 views

### Sets which are unions of translates of each other but aren't single translates

I'm a hobbyist mathematician so any question I ask here might be at risk of closure. I hope this one is good enough, but I'm not sure. This is a continuation of two questions I asked on ...

**16**

votes

**1**answer

736 views

### What makes the amenability of Thompsons group $F$ such a tricky problem?

An open problem that seems to get a lot of attention every once in a while is the amenability of Thompsons group $F$.
The problem seems to generate both proofs and disproofs at a fairly high rate, ...

**6**

votes

**0**answers

89 views

### Checking if a multiplication table represents a group [duplicate]

Given an $n \times n$ multiplication table,
can one check if it represents a group in $o(n^3)$ time?
All properties can be checked by mindless try-all possibilities loops:
Whether there is an ...

**1**

vote

**2**answers

137 views

### Speed and absence of non-constant bounded harmonic functions

For a (symmetric) random walks on countable groups generated by $\mu$, there is a "brute-force computation" argument of Avez (1974) that shows that if the entropy $h_\mu$ is trivial then there are no ...

**5**

votes

**3**answers

249 views

### Can groups of twice-odd order have quaternionic representations?

Let $G$ be a finite group and $\phi\colon G\to \mathrm{GL}_d(\mathbb C)$ be an irreducible representation, with character $\chi$. Recall that
$\phi$ is complex type if $\chi$ is not real-valued,
...

**3**

votes

**0**answers

107 views

### A lemma on verbal conjugacy classes in groups

I'm reading an article whose title is "On Groups With Finite Verbal Conjugacy Classes". Adapting their notations, the authors propose the following lemma.
Let $w$ be a concise word and $G$ a group ...

**3**

votes

**0**answers

111 views

### torsion free for the 2nd cohomology group?

Let $G$ denotes an infinite coutable discrete group with Kazhdan's property (T),
My question is:
is it known that the 2nd cohomology group $H^2(G,\mathbb{Z}G)$ is torsion free?
Thanks in advance!
...

**9**

votes

**2**answers

224 views

### Vertex-primitive graphs with two vertices having almost the same neighbourhood

Hypothesis: Let $\Gamma$ be a vertex-primitive graph with two vertices $u$ and $v$ such that $$|N(u) \cap N(v)|=|N(v)|-1$$
Question: Is it true that $\Gamma$ must either be a complete graph or have ...

**1**

vote

**0**answers

115 views

### Explicitly showing that a free group is LERF [closed]

Let $F$ be a free group on a finite set $X$, and let $M$ be a finitely generated subgroup.
Marshall Hall's theorem states that $M$ is closed in the profinite topology on $F$. That is, $M$ is the ...

**0**

votes

**1**answer

97 views

### All $2$-designs arising from the action of the affine linear group on the field of prime order

Let $p$ be a prime and $\mathbb{Z}_p$ denote as usual the field of order $p$. Let $AL(p)$ be the affine linear group $\{x\mapsto ax+b \;|\; a\in \mathbb{Z}_p\setminus \{0\}, b\in\mathbb{Z}_p\}$. For a ...

**3**

votes

**1**answer

133 views

### Hall subgroups of general linear group

Let $q=p^k$ for some prime $p$, and let $GL_n(\mathbb{F}_q)$ be the group of invertible matrices over the finite field of $q$ elements. If $\pi$ is the set of primes not equal to $p$, does ...

**13**

votes

**1**answer

393 views

### Why is Klein's representation of $PSL_2(\mathbb{F}_7)$ hard to obtain?

In his famous article [1] Klein constructs a representation of $G=PSL_2(\mathbb{F}_7)$ in $\mathbb{C}^3$ (of which the first invariant polynomial of three variables gives rise to the famous Klein's ...

**10**

votes

**3**answers

444 views

### Natural associative law for a ternary “group”?

Suppose one were to define a group-like structure based on a set $G$
with a ternary (rather than binary) operator $g( a, b, c ) = \left< a, b, c \right>$.
One possible definition for the ...

**2**

votes

**3**answers

381 views

### Lucido's three prime lemma

Let G be a finite solvable group. If p,q,r are distinct primes dividing |G|, then G contains an element of order the product of two of these three primes.
This is lucido's three prime lemma. I ...

**3**

votes

**0**answers

72 views

### A Karrass-Solitar or Ivanov-Schupp for profinite groups

Let $F$ be a nonabelian free profinite group, $H \leq_c F$ finitely generated with $[F:H] = \infty$. Must there be some $\{1\} \neq N \lhd_c F$ such that $N \cap H = \{1\}$?

**0**

votes

**1**answer

90 views

### on lifting extensions

Let $G$ be a connected reductive group with $G_{der}$ simply connected and $T$ a maximal torus over an algebraically field $k$.
We consider a extension $\tilde{T}$ of the maximal torus $T$ by a torus ...

**7**

votes

**1**answer

260 views

### New relator in hurwitz group

I have found that $([a,b]^2[a,b^2])^n$ is a good relator to use in my search for quotients of $G := \langle a, b \ | \ a^2, b^3, (ab)^7, [a,b]^{10} \rangle$. For n<=5 $H := \langle a, b \ | \ a^2, ...

**6**

votes

**0**answers

94 views

### Outer automorphisms of direct product of residually finite groups

Let $A,B$ be finite generated residually finite groups such that $\mathop{Out}(A)$ and $\mathop{Out}(B)$ are residually finite.
Is $\mathop{Out}(A\times B)$ residually finite?
If not, what is the ...