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

**6**

votes

**1**answer

212 views

### Graph with group structure?

Is there any established theory of graphs which themselves are groups? I don't mean Cayley graphs or "graphs of groups". I mean a graph whose set of vertices forms a group, where the group operation ...

**2**

votes

**1**answer

156 views

### Reference sought regarding the fact that all non-abelian finite simple groups are 2-generated

I've learned that all non-abelian finite simple groups are $2$-generated,
i.e. have a generating set of cardinality $2$.
Is there a reference to this statement which does not just point to the
...

**3**

votes

**0**answers

129 views

### Which Dihedral Groups are $\text{CI}$-Groups?

Let $D_{n}$ denotes the dihedral group of order $2n$. Firstly, for self-referencing of the question, I give some definitions which are standard.
Let $G$ be a finite group. A subset $S$ of group $G$ ...

**0**

votes

**0**answers

55 views

### “4th order” floretions- cyclic transformation question

In response to the last paragraph mentioning "swapping" operations in this post, I would like to mention what the reference is to and one question I currently have.
Assume $X = abCD$ is some 4th ...

**1**

vote

**1**answer

130 views

### Perfect centerless normal subgroups

Let $S$ be a non-trivial simple group and suppose $S \trianglelefteq G$ if $C_G(S)=1$ then $S$ is characteristic in $G$. To prove this let $\phi$ be an automorphism of $G$ and note that the ...

**11**

votes

**1**answer

198 views

### $\pi_1$ of the moduli of G-bundles on elliptic curves and the double affine braid group

For a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$, the fundamental group of $\mathfrak{h}_\text{reg}/W$ (where $\mathfrak{h}$ is the Cartan subalgebra, $\mathfrak{h}_\text{reg}$ is the subset ...

**2**

votes

**0**answers

65 views

### maximal elementary abelian p-subgroups of finite groups of Lie type

Let $G$ be a simple algebraic group over an algebraically closed field k of characteristic $p$. Let $G(\mathbb{F}_{p^r})$ be the corresponding finite group of Lie type. Is it true that every maximal ...

**4**

votes

**1**answer

92 views

### Maximal cyclic quotient of a $p$-group

Let $G$ be a finite abelian $p$-group, $p$ a prime. I say that a pair $(G',\varphi)$ is a maximal cyclic quotient (please excuse me if this definition already exists and refers to a different concept) ...

**13**

votes

**1**answer

191 views

### Free generators for the fat commutator subgroup

There is a homomorphism $\langle x,y\rangle\to\langle x\rangle$ of free groups, sending $y$ to $1$. We can combine this with the other obvious homomorphism to get a surjective homomorphism
$$ ...

**1**

vote

**0**answers

66 views

### associativity of the extension of finie groups [closed]

Following my previous question I have two questions:
1.the extensions of the group is associative or not, i.e. as we know by the notations of atlas if $S={\rm PSL}(2,q)$, where $q=p^f>9$, then ...

**7**

votes

**0**answers

92 views

### Number of matrices with 'small' entries in integer matrix groups

Let $n$ be a positive integer, and let $G \leq {\rm GL}(n,\mathbb{Z})$.
Given a bound $b \in \mathbb{N}$, let $e_b$ be the number of elements
of $G$ all of whose matrix entries have absolute value ...

**3**

votes

**0**answers

80 views

### Profinite topology on free metabelian group

Let $M$ be free metabelian group of rank $n$. By work of Coulbois, M is $LERF$ and is not $RZ_2$, that means every finitely generated subgroup of $M$ is closed in the profinite topology of $M$ but the ...

**7**

votes

**1**answer

166 views

### Groups without property (T) but all finite quotients are expanders

What is an example of a group $G$ which
1- is finitely generated by $S$,
2- does not have property (T),
3- admits infinitely many finite quotients which do not factor through an homomorphism $G ...

**4**

votes

**1**answer

126 views

### Extensions of $\Bbb Z_3$ by $PGL(2,q)$ where $q$ is odd

Let $q$ be odd. If $G$ is a finite group such that $G$ has a normal subgroup $H$ of order $3$ such that $G/H\cong {\rm PGL}(2,q)$, what can we say about $G$. Is it true in general that $G\cong {\Bbb ...

**7**

votes

**0**answers

155 views

### A conjecture of Lubotzky on ranks of subgroups of special linear groups over the integers

In a 1985 paper named "Dimension function for discrete groups" Lubotzky conjectured that:
For any integer $n \geq 3$ the group $\mathrm{SL}_n(\mathbb{Z})$
contains infinitely many finite index ...

**5**

votes

**2**answers

301 views

### Inverse Galois problem for simple Lie type groups

Progress towards the Inverse Galois problem over $\mathbb{Q}$ is very well documented for sporadic groups ($M_{23}$ is the only case open) and for $PSL_n(q)$ (a lot of cases known, but wide open in ...

**4**

votes

**0**answers

99 views

### Exotic 2-adic lifts of mod $2$ Steinberg idempotent

Denote $B_n$ the Borel subgroup of $Gl_n(Z/2)$, i.e., the subgroup of
upper triangular matrices, $\Sigma _n$ the subgroup of permutation matrices.
The (conjugate) Steinberg idempotent is defined to be ...

**5**

votes

**1**answer

118 views

### To calculate $Tor_1^G(\mathbb{Z},N_{ab})$ and $Tor_1^Q(\mathbb{Z},N_{ab})$

Let $G$ be a finite $p$-group and $N$ be a normal subgroup of $G$. I wish to calculate $Tor_1^G(\mathbb{Z},N_{ab})$ and $Tor_1^Q(\mathbb{Z},N_{ab})$, where $Q=G/N$. I could not found any lecture notes ...

**2**

votes

**1**answer

109 views

### k-fellow traveler property and automatic structur

Let P be a permutation group with some generating set S and let W be the word acceptor automaton of P, if I know the value of k (k-fellow-traveller property of CayleyGraph CG(P,S)).
I realized that ...

**0**

votes

**0**answers

76 views

### Abelian centralizer groups (CA-groups)

I am searching for all information about CA-groups [abelian centralizer groups] and i just found a German book [Huppert] and Nilpotent Centralizer group of Suzuki in 44 pages and Group theory book of ...

**3**

votes

**0**answers

90 views

### Schur Multiplier of Tarski Monsters

Is it known whether the Schur Multiplier of the Tarski monsters are finitely generated?

**8**

votes

**0**answers

150 views

### A meager subgroup of the real line, which cannot be covered by countably many closed subsets of measure zero?

Is there a ZFC-example of a subgroup $H$ of the real line $\mathbb R$ such $H$ is meager, has zero Lebesgue measure, but cannot be covered by countably many closed subsets of measure zero in $\mathbb ...

**3**

votes

**2**answers

128 views

### On the Magnus Representation of Free Metabelian Group

Let $F=\langle x_1,x_2\rangle$ be a free group of rank $2$ and $\Phi=F/F''=\langle \overline{x}_1, \overline{x}_2\rangle$ where $F''$ is second derived subgroup of $F$ (i.e. $F'=[F,F]$ and ...

**7**

votes

**2**answers

385 views

### asymptotic for the number of involutions in GL(n,2)

Is it known how the number of involutions in $GL_n(2)$, the group of $n\times n$ matrices over $\mathbb{Z}/2\mathbb{Z}$, behaves as $n\to\infty$ ?
Equivalently, one may ask this for the number of ...

**2**

votes

**1**answer

74 views

### The maximal possible rank of a subgroup of a product of special linear groups

In this question I ask for a generalization of What is the maximal possible rank of a subgroup of a special linear group mod a prime?
Let $p_1, \dots, p_r$ be $r$ distinct odd primes.
Set $$G = ...

**7**

votes

**1**answer

97 views

### What is the maximal possible rank of a subgroup of a special linear group mod a prime?

Let $p$ be a prime number, and let $\mathbb{F}_p$ be the unique field of cardinality $p$.
What is $\max \{d(H) : H \leq \mathrm{SL}_3(\mathbb{F}_p)\}$?
Here we denote by $d(G)$ the smallest ...

**11**

votes

**1**answer

282 views

### Elementary consequences of commuting limits and colimits over groups

This is a crosspost of this MSE question.
In this n-cat cafe post, it is proven that for finite groups $G,H$ of coprime order, $G$-colimits and $H$-limits commute. Later on the following theorem is ...

**5**

votes

**1**answer

162 views

### Maximal subgroups of special linear groups over finite fields

Let $p$ be a prime number, and denote by $\mathbb{F}_p$ the field with $p$ elements.
Is there a classification of the maximal subgroups of $G = \mathrm{SL}_3(\mathbb{F}_p)$ ?
I am interested in ...

**9**

votes

**0**answers

128 views

### Inducing up the group homomorphism between mapping class groups

There are many ways to embed the braid group into the mapping class group of a surface. To describe one of them, let ${C}_{2g+2}(\mathbb{D}^2)$ be the configuration of unordered $2g+2$ points in the ...

**14**

votes

**2**answers

577 views

### Braid group on 4 strands

Consider the braid group $B_4$ with generators
$a=\ $,
$b=\ $
and $c=\ $
Assume
$$\alpha, \alpha'
\in
\langle a,c\rangle{\smallsetminus}(\langle c\rangle{\cdot}\langle a\rangle{\cdot}\langle ...

**2**

votes

**0**answers

131 views

### calculation in a group ring

I have some problems with the verification of the third equation in Lemma 1 in this paper.
First of all, one has to notice that there is at least one Error in the Definition of $a_{\kappa,\nu}$ ...

**4**

votes

**1**answer

128 views

### Centralizer of hermitian matrices with zero trace

In Quantum Physics one often has to deal with commutators.
Here I want to denote by $H_0$ the set of all hermitian matrices with trace equal to zero!
One can easily relate it to ...

**5**

votes

**0**answers

68 views

### Normalizer of a reflection supgroup of a finite complex reflection group

Let $W \leq \operatorname{GL}(\mathbb{C}^\ell)$ be a finite complex reflection group acting on the finite dimensional complex vectorspace $\mathbb{C}^\ell$.
Let $W' \leq W$ be a reflection subgroup, ...

**4**

votes

**0**answers

216 views

### a characterization for special linear group

Thanks for any help or comments.
In my research I need a characterization for special linear groups over finite fields by some information of its subgroups, especially centralizers. I saw the ...

**4**

votes

**1**answer

90 views

### Computations with conetypes of hyperbolic groups

I'd like to know if there exists (and, in this case, where I can find it) some computer program/programming language/any kind of software that can find explicitly the conetypes of a hyperbolic group ...

**0**

votes

**0**answers

46 views

### The name of a class of linearly ordered groups

My friend asked me to ask his question here. Is there a special name for a linearly ordered group $G$ such that for every positive element $g\in G$ there exists an element $h\in G$ such that ...

**4**

votes

**0**answers

135 views

### Is the Tensor/Exterior square $G\otimes G$ or $G\wedge G$ of infinite p-group also a p-group?

Let $G$ be an infinite countable p-group. Is it true that $G\otimes G$ or $G\wedge G$ are also p-groups? (where G acts on itself by conjugation). For simplicity, you can consider that $G=[G,G]$, and ...

**3**

votes

**1**answer

115 views

### extending homomorphisms to semidirect products

Let $A = F_2$ be the free group on two generators (not sure if this is important.)
Suppose you have a semidirect product $A\rtimes C$ coming from some homomorphism $\beta: C\rightarrow ...

**7**

votes

**4**answers

571 views

### fixed points of permutation groups

As is well-known (see, for example, a nice exposition by our own Qiaochu: https://qchu.wordpress.com/2012/11/07/fixed-points-of-random-permutations/) that the distribution of the number of fixed ...

**5**

votes

**0**answers

80 views

### Maximum relator and hyperbolicity

It is a well-known theorem that a finitely generated group is hyperbolic if and only if it admits a finite Dehn presentation. To prove the "only if" direction one proceeds roughly as follows:
Suppose ...

**3**

votes

**0**answers

119 views

### Nilpotent Groups of Generalized Prime Exponent

A group $G$ is called of generalized exponent $n$ if there exists elements $a_1,\dots,a_n \in G$ such that $x^{a_1}\cdots x^{a_n}=1$ for all $x\in G$, where $x^a=a^{-1}xa$.
See the following question
...

**5**

votes

**1**answer

132 views

### Minimal number of algebra generators of a group ring

Let $k$ be a commutative ring with unit and let $(A,\varepsilon)$ be a (not necessarily commutative) augmented, finitely generated $k$-algebra with augmentation ideal $I$.
If $\mu(A)$ denotes the ...

**4**

votes

**0**answers

76 views

### Sharp isoperimetry in the discrete Heisenberg group

The exact shape of the set which has the best isoperimetry in the continuous Heisenberg is (from what I know) a difficult open problem. This brought to wonder what is known in the discrete case?
More ...

**9**

votes

**0**answers

159 views

### Is an inclusion of finite groups with boolean lattice, linearly primitive?

Let $(H \subset G)$ be an inclusion of finite groups.
Definition: Let $W$ be a representation of $G$, and $X$ a subspace of $W$.
Let the fixed-point subspace $W^{H}:=\{w \in W \ \vert \ kw=w \ ...

**5**

votes

**1**answer

203 views

### Is there a one relator group with property (T)?

Is there an $n > 2$, and some $x \in F_n$ (the free group on $n$ generators) such that the quotient of $F_n$ by the normal subgroup generated by $x$ has Kazhdan's property $\mathrm{(T)}$ ?

**3**

votes

**1**answer

236 views

### Maximal ideal of group ring

Let $R$ be a finite commutative ring with identity and $G$ an finite abelian group. Is there any more conditions (on $R$ or on $G $) under which we can characterize maximal ideals of group ring $RG $, ...

**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

237 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 $f: H\to GL_m(\mathbb{Z})$ ...

**33**

votes

**0**answers

945 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 $A \not\cong A \oplus \mathbb{Z}$?