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

**0**

votes

**0**answers

27 views

### Lexicographic rank of an odd/even permutation

Let $S_n$ be the set of all permutations of integers from $1$ to $n$. Let $P_1$ and $P_2$ be two partitions defined on $S_n$ as follows.
$P_1$ is the set of all those permutations which have even ...

**7**

votes

**0**answers

89 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 ...

**4**

votes

**2**answers

254 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

75 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 ...

**2**

votes

**0**answers

46 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

85 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

58 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

73 views

### Schur Multiplier of Tarski Monsters

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

**7**

votes

**0**answers

119 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 ...

**2**

votes

**2**answers

107 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

**1**answer

190 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

70 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 = ...

**-1**

votes

**0**answers

109 views

### Abelian quotient of a surface group

Let $S$ be a compact closed surface of genus two. Let $x_1$, $y_1$, $x_2$, and $y_2$ be the standard generating set for $\pi_1(S)$ (i.e. $\pi_1(S) \ = \ <x_1, y_1, x_2, y_2 | [x_1,y_1][x_2,y_2] = ...

**7**

votes

**1**answer

92 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 ...

**8**

votes

**0**answers

145 views

+50

### 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

149 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 ...

**8**

votes

**0**answers

106 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 ...

**-2**

votes

**0**answers

66 views

### How to find composition series for $GL_{2}(\mathbb{Z}/5\mathbb{Z})$ and $GL_{2}(\mathbb{Z}/25\mathbb{Z})$ [migrated]

I'm having trouble trying to find composition series for $GL_{2}(\mathbb{Z}/5\mathbb{Z})$ and $GL_{2}(\mathbb{Z}/25\mathbb{Z})$. I have no idea how to construct this... Someone says that the number ...

**12**

votes

**2**answers

478 views

+50

### 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\backslash(\langle c\rangle{\cdot}\langle a\rangle{\cdot}\langle ...

**2**

votes

**0**answers

122 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 $a_{\kappa,\nu}$ ...

**3**

votes

**1**answer

121 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 ...

**3**

votes

**0**answers

50 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, ...

**-2**

votes

**0**answers

102 views

### Free abelian subgroups of SL(3,$\mathbb{Z}$) [migrated]

Does SL(3,$\mathbb{Z}$) have any free abelian subgroup of rank > 2?
I want to find 3 $\times$ 3 integer matrices with determinant 1 such that the matrices are commutative, but there exists no other ...

**4**

votes

**0**answers

197 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

81 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

40 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

114 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

108 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

540 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

74 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

113 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

130 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

73 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 ...

**8**

votes

**0**answers

63 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

190 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

218 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

55 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

164 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?

**8**

votes

**1**answer

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

**32**

votes

**0**answers

857 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

503 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

70 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 ...

**-1**

votes

**0**answers

45 views

### Question on uniquely $p$-divisible groups [migrated]

I know that an abelian group is uniquely (i.e. torsion-free) divisible iff is has a natural structure of a $\mathbb{Q}$-vector space. Is it true that an abelian group is uniquely $p$-divisible iff it ...

**4**

votes

**0**answers

72 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

341 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

308 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

302 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

150 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

89 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

163 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 ...