# Tagged Questions

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

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**1**answer

131 views

### Some question on haar measure for sumsets of closed subsets of profinite groups

Let $H$ be a profinite group with the haar measure $\mu_H$. Let $H_1$ and $H_2$ be closed subgroups of $H$. $H_1$ and $H_2$ have their own haar measures $\mu_{H_1}$ and $\mu_{H_2}$ respectively.
...

**10**

votes

**1**answer

196 views

### commutators in upper triangular matrices

Consider the group $T_p(n)$ of all non-singular upper triangular matrices with entries in $\mathbb{F}_p.$ Its commutator subgroup is $U_p(n)$ (all elements in $T_p(n)$ with $1$s on the main diagonal). ...

**4**

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**1**answer

195 views

### Does there exist finite dimensional irreducible representation of Euclidean or Poincare group in which translation and rotation both act nontrivially?

Does there exist any finite dimensional irreducible rep. of Euclidean or Poincare group in which translation and rotation both act nontrivially?
Let me firstly clarify my question. For example, we ...

**4**

votes

**0**answers

133 views

### Lifting automorphisms of quotient groups

I am concerning here a natural question:
Problem: Let $G$ be a finite group, and let $N$ be a characteristic subgroup of $G$. When can an automorphism $\varphi\in\mathrm{Aut}(G/N)$ be lifted to an ...

**3**

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147 views

### The homotopy type of the mapping space $Map_{B\rho}(BS^1,BG)$? for $G$ a compact Lie group

Given a homomorphism $\rho:S^1\rightarrow G$ with $G$ a compact Lie group there is an induced map of classifying spaces $B\rho:BS^1\rightarrow BG$. What is known about the homotopy type of the mapping ...

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**0**answers

242 views

### How bad can $SK_1$ of a commutative ring be?

For a commutative ring $R$ define $\mathrm{SK}_1(n, R)=\mathrm{SL}(n, R)/\mathrm{E}(n, R)$, the quotient of the special linear group by its subgroup generated by the elementary matrices. When $n\...

**7**

votes

**1**answer

256 views

### Generic set that is a proper subgroup

For a group $G$ generated by a finite set $S$ we denote by $B_{G,S}(n)$ the ball of radius $n$, that is the set of all elements in $G$ which are expressible as products $x_1x_2\ldots x_n$ where $x_i\...

**2**

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**1**answer

188 views

### Cohomologically trivial $G$-modules

Is there a finite non-abelian $2$-group $G$ without non-trivial elementary abelian direct factor and of order $2^9$ satisfying the following condition: $$Z(G) \cap Z(\Phi(G))= \langle \prod_{i=1}^{2^d}...

**4**

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**0**answers

70 views

### Abnormal subnormal series

Consider a group $G.$ Is it possible for $G$ to have a subnormal series $G \triangleright G_1 \triangleright \dotsc \triangleright G_n \triangleright \dotsc$ which cycles - that is, with $G_{i+k} \...

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votes

**2**answers

127 views

### Finite solvable groups with abelian Fitting subgroup

Let $G$ be any finite solvable group with Fitting subgroup
$F(G)$. Which conditions on $F(G)$ makes $G$ to be supersolvable?
(It is well-known that any finite solvable group with cyclic Fitting ...

**15**

votes

**5**answers

487 views

### Cayley graphs of $A_n.$

Consider the Cayley graphs of $A_n,$ with respect to the generating set of all $3$-cycles. Their properties must be quite well-known, but sadly not to me. For example: what is its diameter? Is it an ...

**3**

votes

**1**answer

197 views

### Braid groups over the sphere

Can any one give me an example of surjective homomorphism on braid groups on the sphere that is not injective?
Such that $B_{n}(S^2)$ is generated by $\sigma_1,\sigma_2, \dots, \sigma_{n-1}$ which are ...

**2**

votes

**1**answer

171 views

### 2-closure of a permutation group

Let $G$ be a group acting on a set $\Omega$ faithfully. Then 2-closure of $G$ denoted by
$G^{(2)}$ is the largest subgroup of the symmetric group of $\Omega$ with
the same orbits as $G$ on $\Omega\...

**6**

votes

**1**answer

331 views

### Liouville property - a very basic question

Let $\mathbb{F}_2$ be the free group on two generators. By a result of Kaimanovich and Vershik, for each measure $\mu$ on $\mathbb{F}_2$ such that the support of $\mu$ generates $\mathbb{F}_2$, we ...

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**0**answers

190 views

### Unique product groups (and semigroups)

A group $G$ is called a u.p.-group (short for unique product group) if for all nonempty finite subsets $A,B\subseteq G$, there exists an element $g\in A \cdot B$ which can be uniquely written as a ...

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votes

**2**answers

188 views

### Finding an “optimal” quotient in a free group

Consider the abelian free group $G = \mathbf{Z}^n$ of rank $n$ and a finite subset $A \subset G \setminus \{0\}$. Since $G$ is residually finite, there is a subgroup $H \subset G$ such that $A \cap H =...

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**3**answers

242 views

### Signs in Chevalley's commutator formula

I am trying to understand presentations of twisted groups of Lie type (specifically $^2D_5$) over finite fields using Steinberg's presentations (for instance from Gorenstein, Lyons and Solomon, Number ...

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**0**answers

132 views

### Groups with infinitely many finite conjugacy classes

I've been coming across the condition "IMFCC: having infinitely many finite conjugacy classes" often in recent times and I was wondering if there is any serious difference between having "IMFCC" and ...

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votes

**1**answer

103 views

### Intransitive finite irreducible linear groups whose orbits are all large

I am interested in intransitive irreducible linear subgroups $G\subseteq\mathrm{GL}_n(\mathbb{F}_p)$ acting on $V-\{0\}=\mathbb{F}_p^n-\{0\}$ in the natural way, such that all of the orbits are very ...

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258 views

### Cardinality of factors of infinite non-abelian groups

Let $A$ and $B$ be arbitrary nonempty subsets of a group $G$. Then
the product $AB$ is called direct, and we denot it by $A \cdot B$,
if the representation of every its element by $x=ab$ with $a\in ...

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**0**answers

295 views

### A question concerning model theory of groups

Several days ago, Professor Martin Bridson gave a very nice talk in my department. Some questions concerning his talk came into my brain Since I am neither a model theorist nor a algebraist, I am not ...

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161 views

### Detecting a module for the free group algebra on a finite quotient

Let $F_2$ be the free group on two generators $x,y$ and let $R$ be the group algebra $\mathbf{Q}[F_2]$. Let $a,b,c$ be integers. Then define a right $R$-module
$M = R / (ax + by + c) R$.
I am ...

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**2**answers

269 views

### Thompson group $V$

R. Thompson introduced three groups $F\subset T\subset V$. The question concerning amenability of $F$ is still unanswered and has attracted much attention. I have read that Thompson group $V$ contains ...

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**1**answer

339 views

### Cohomology of central extensions of groups

Let G be a central extension of a finite group H by $Z/2$. I need an explicit description of the differentials $d_2$ and $d_3$ in the Lyndon-Hochschild-Serre spectral sequence which converges to the ...

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votes

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96 views

### How to enumerate a discrete group of matrices by their Frobenius norm?

Suppose I have a discrete group $G<\mathrm{SL}_2(\mathbb{C})$,
and it is finitely generated by some known generators.
That is, $G=\langle g_1,\dots,g_n\rangle$.
The Frobenius norm of a matrix $m=\...

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vote

**1**answer

190 views

### What are the finite subgroups of $\operatorname{Sp}_{2n}(\mathbb{Z})$?

I've read the following question:
Finite subgroups of ${\rm SL}_2(\mathbb{Z})$ (reference request)
and it made me wonder. It's easy to see that $\operatorname{SL}_2(\mathbb{Z})=\operatorname{Sp}_2(\...

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votes

**1**answer

158 views

### Braid group: Can a left-twist increase the number of right twists?

Disclaimer: This question was first posted on math.se without any answer.
This is something that naturally occurs in my research, but I am no expert on this - it feels like a natural question so I am ...

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votes

**1**answer

129 views

### The relationship between $p$-solvable Group and solvable group [closed]

Can anyone please tell me The relationship between $p$-solvable Group
and solvable group.and find an example of a $p$-solvable group that is not solvable group or vice-versa.

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229 views

### A “direct” proof that hyperbolic groups are not amenable

I am looking for a proof that a finitely generated hyperbolic groups is non-amenable [unless it is virtually cyclic] which is as "metric/combinatoric" as possible.
Here are the two proofs I am aware ...

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**1**answer

118 views

### A decomposition of $w_0$ which is similar to the reduced decomposition

Some basic definitions about reduced decomposition:
In the symmetric group $S_n$, let $s_i$ denote the adjacent transposition $(i,i+1),i\in \{1,2,\cdots,n-1\}.$ Since $S_n$ is generated by adjacent ...

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**1**answer

177 views

### A question on the commutativity degree of the monoid of subsets of a finite group

The commutativity degree $d(G)$ of a finite group $G$ is defined as the ratio
$$\frac{|\{(x,y)\in G^2 | xy=yx\}|}{|G|^2}.$$It is well known that $d(G)\leq5/8$ for any finite non-abelian group $G$. If $...

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vote

**2**answers

109 views

### Any analysis on phase of eigenvalue of unitary matrix?

I understand that there are invariant Haar measure for eigenvalues of unitary matrix. I further understand that absolute value of eigenvalues of unitary matrix is 1. But, I could not find any analysis ...

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votes

**2**answers

527 views

### Is the absolute Galois group the same as the automorphism group? [closed]

Is the absolute Galois group $\mathrm{Gal}(\overline{\mathbf{Q}}|\mathbf{Q})$ the same as the group $\mathrm{Aut}_{\mathbf{Q}}(\overline{\mathbf{Q}})$ the automorphism group in the category of $\...

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votes

**2**answers

440 views

### For a ring R, does $GL_n(R)$ embed into $GL_m(F)$ for some field F?

Suppose $R$ is a noncommutative ring. What is the sufficient and necessary condition for $GL_n(R)$ embedding into $GL_m(F)$ for some field $F$?
In particular, what if $R$ is the group ring $\mathbb{Z}...

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79 views

### Embedding a cancellative monoid into another in such a way that $|X-x|=|X|$, where $X$ is a fixed finite set and $x\in X$

Preliminaries.
Let $\mathbb A = (A, +)$ be a possibly non-commutative semigroup. For $X, Y \subseteq A$ we set
$$
X - Y := \{a \in A: a + y \in X\text{ for some }y \in Y\},
$$
which is just the usual ...

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votes

**1**answer

99 views

### free subgroups of the fundamental group of nonorientable surfaces

Sorry for this possibly trivial or stupid question, but I'm very far from being an expert in Algebra.
Let G be the fundamental group of the nonorientable surface of even rank n=2k (n generators, 2n ...

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votes

**1**answer

198 views

### Embedding a cancellative monoid into another in such a way that a prescribed element becomes left-invertible

Let $\mathbb A = (A, +_A)$ be a cancellative, but possibly non-commutative, monoid with identity $0$, and fix an element $x \in A$. Does there always exist a cancellative monoid $\mathbb B = (B, +)$ ...

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votes

**1**answer

244 views

### Does the Fano plane “embed” in the complex projective plane?

$PSL(2,7)$ acts on the projective plane over $\mathbb{F}_2$ (the Fano plane) through its identification with $GL(3,2)$. It also acts on the projective plane over $\mathbb{C}$ through either of its ...

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vote

**0**answers

125 views

### Which conjectures are proved for sofic groups? [closed]

Which conjectures about groups are resolved in case of sofic groups?
I know two examples:
Kaplansky's direct finiteness conjecture (proved by Gabor Elek).
Some versions of Ornstein's isomorphism ...

**31**

votes

**1**answer

568 views

### Identities of commutators

Let $G$ be a group and set $[x,y]:= x^{-1}y^{-1}xy$ as usual and consider it as a binary operation.
Question: Is there a description of the identities that the operation $[.,.]$ satisfies for all ...

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vote

**0**answers

77 views

### Infinite Hirsch length [closed]

Can a residually finite group $G \in LFin \rtimes VPoly$ ($G$ is a semi-direct product of a locally finite group by a virtually polycyclic group) have an infinite Hirsch length?

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178 views

### Mixed up by definitions of mildly mixing

Here are two setup where the notion of "mildly mixing" comes up: for representations and for group acting by measure preserving transformations (see definitions below). Since a natural class of ...

**9**

votes

**1**answer

213 views

### p-groups such that the center is contained in many cyclic subgroups

I'm looking for examples of $p$-groups $G$ with the following three properties:
the center of $G$ is $\mathbb{Z}/p$, and
$G^{\text{ab}} = (\mathbb{Z}/p)^n$ for some $n$, and
for every $g \in G$ ...

**0**

votes

**0**answers

67 views

### Matrix representation of the Heisenberg quaternionic group

Equiped with the law $(a,b,c)\circ (a',b',c') = (a + a', b + b', c + c' + ab'),$ the matrix representation of the Heisenberg group $H^3= \mathbb C\times \mathbb R$ is given by
$$
\begin{pmatrix}
1 &...

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votes

**3**answers

708 views

### Simplicity of $A_n.$

I am teaching an introductory group theory course, and it has come to the inevitable proof that $A_n$ is simple for $n\geq 5.$ Now, there seem to be a number of proofs that I can find - one the "...

**0**

votes

**0**answers

110 views

### On the structure of groups according to their conjugacy classes

A finite group $G$ satisfies property $P_n$ if for every prime integer $p$, $G$ has at most $(n-1)$ non-central conjugacy classes the order of the representative element of which is a multiple of $p$. ...

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121 views

### Cocycle condition for 2-groups

I know that if $\omega_d(g_1, \ldots, g_d)$ is an d-cocycle characterized by $H^d(G,U(1))$, it satisfies the co-cycle condition
$(d\omega_d)(g_1, \ldots, g_{d+1}) = g_1.\omega_d(g_2,\ldots,g_{d+1}) + ...

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vote

**0**answers

92 views

### Exponent of the quotient of the commutator of a free group

Let $F$ be a free group on two generators, let $F^p$ denote the normal subgroup of $F$ generated by the $p$-th power, $p$ a prime. Do we know anything about the exponent of $[F,F]/[F^p,F]$.
Edit: $G=...

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114 views

### Mal'cev completions of finitely generated torsion-free nilpotent groups

There is some question from geometric group theory:
One wonders if the following conditions are equivalent for finitely generated torsion-free nilpotent groups $\Gamma$ and $\Lambda$:
$\Gamma$ and $\...

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votes

**0**answers

67 views

### Have locally principal crossed homomorphisms been studied?

Take a (multiplicative finite) group $H$ acting on the left (by automorphisms) on an (additive finite) abelian group $A$, and recall that the abelian (additive) group of crossed homomorphisms from $H$ ...