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

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### Presentation of hyperbolic groups [closed]

Is it true that all hyperbolic groups are finitely presented? If yes, what is the right reference for that?

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217 views

### Normal subgroups of Aut(M)

Let $S$ be the set of all finite permutations of $\mathbb{N}$, i.e. they fix all but a finite set, and $A\subset S$ the set of all even permutations.
Theorem The normal subgroups of $S_\infty$ are ...

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197 views

### Closed subgroups of $\mathrm{SO}(4)$

My question is quite simple : we know all closed subgroups of $\mathrm{SO}(3)$; is it also known what are the closed subgroups of $\mathrm{SO}(4)$?

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199 views

### Show that $SL_2(\mathbb{F}_p)$ is quasi-random

Terry Tao gives this oblique definition of quasirandom group in his notes 3
$G$ is quasi-random (of order $D$) if all non-trivial unitary representations $\rho: G \to U(H)$ have dimension at ...

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427 views

### Kazhdan's property (T) vs. residual finiteness

I have asked this question already on mathstackexchange but got no answer (see http://math.stackexchange.com/questions/1795795/kazhdans-property-t-vs-residual-finiteness) and it was suggested that I ...

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96 views

### Orthogonality relations for unitary representations of infinite (finitely generated) groups

Let $G$ be a group, and consider the matrix elements of finite dimensional irreducible unitary representations of $G$ over $\mathbb{C}$ as functions $f:G\to \mathbb{C}$. If $G$ is finite, any two ...

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69 views

### Cubic Directed Cayley graphs of 2-generated torsion-free groups

Is there a torsion-free group $G=\langle x,y \rangle$ such that the directed Cayley graph $\Gamma=Cay(G,\{x,y,x^{-1}y\})$ contains a finite cubic induced subgraph? The vertex set of $\Gamma$ is $G$ ...

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485 views

### On progress towards inverse Galois problem over rationals

I have heard that 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 $PSL_n(q)$.
From where I can read ...

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1k views

### What's the supersymmetric analogue of the Monster group?

Bosonic string theory lives in 26 dimensions, and it gives a conformal field theory where the field is a map from a Riemann surface to $\mathbb{R}^{24}$. The Leech lattice $L$ is an even unimodular ...

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269 views

### Does the injection $\text{Aut}(F_n) \hookrightarrow \text{Aut}(F_{n+1})$ split?

Let $F_n$ be the free group on $n$ letters.
The question is as in the title: letting $i:\text{Aut}(F_n) \hookrightarrow \text{Aut}(F_{n+1})$ be the natural injection, does there exist a homomorphism $...

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87 views

### Computing the order of elements in a non abelian exterior square of a finite group

If we have an explicit group $G$, and we pick two elements $g,h \in G$, could we find the order of the element $g \wedge h \in G \wedge G$?
The best thing I could find is Theorem 1.1 in Ellis' Book (...

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37 views

### The Socle of locally nilpotent $p$-group infinte rank

The Socle (is the subgroup generated by the minimal normal subgroups of $G$) of abelian $p$-group of infinite rank has infinite rank. (The term “rank” in the sense of the Mal’cev special or ...

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91 views

### Primary invariants

This question is related to the earlier question which is in the given link:
Primary invariants of a finite group
Let $G$ be a finite group and $V$ a complex representation of degree $n$, and let $...

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122 views

### A more precise description of conjugation of semi-simple subgroups

Let $G$ be a semi-simple algebraic group over $\mathbb{Q}$, I would like to find an integer $d>0$ only depending on $G$ with the following property. For any two semi-simple $\mathbb{Q}$-subgroups $...

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115 views

### An infinite torsion group $G$ with finite type $K(G,1)$?

There is a famous open problem in group theory that asks:
Does there exist an infinite finitely presented torsion group?
The general belief being that such groups exist. I would like to know ...

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117 views

### About the Eigenvalues of Orthogonal Matrix plus Perturbation

Let $O$ be an orthogonal matrix, $O^T O = I$, thus its eigenvalues lie on the unit circle, $\lambda(O)=e^{i\theta}$. Furthermore, assume the form
$O = X Y$, where both matrices satisfy $X^2 = I$ and $...

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243 views

### About the conjugation of semi-simple subgroups

Let $G$ be a semi-simple algebraic group over $\mathbb{Q}$, I would like to find an integer $d>0$ only depending on $G$ with the following property. For any two semi-simple $\mathbb{Q}$-subgroups $...

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39 views

### Explicit description of fields with ramification conditions

Let us fix an algebraically closed field $k$ of characteristic 0. If I understood correctly, the Riemann Existence theorem guarantees us existence of the field (Galois-)extension, say $F$, of $k(t)$ ...

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97 views

### Chief factors and local formation

Every thing below is concerned with finite groups.
My question is about this paper
A class of groups is a collection $\mathcal{X}$ of groups with the property that
if $G \in \mathcal{X}$ and if $H \...

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116 views

### Is there a matrix representation of the permutation group whose character is the Markov trace?

Let $g$ be an element in the permutation group (symmetric group) $S_N$. Define the Markov trace of $g$ (denoted $\text{tr}_k g$) as
$$\text{tr}_kg = k^\text{number of cycles in $g$} ,$$
which depends ...

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296 views

### Finite groups inside an infinite group with the same homology

Suppose we have a triple of groups $G,H,K$ verifying the follwing conditions
$G$ and $H$ are finite groups and $K$ an infinite group.
there exists two monomorphisms $G\rightarrow K\leftarrow H$ ...

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359 views

### Verify that a group is hyperbolic via computer algebra

I would like to know whether there is some computer algebra software that can be used to verify if a group, given by a finite presentation, is hyperbolic (in the sense that it terminates with "yes" if ...

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### Computing characters of $\alpha$-projective representations

Given a finite group $G$, a finite cyclic group $A$ (viewed as a subgroup of $\mathbb{C}^{\times}$, i.e. generated by a $|A|$-th root of unity), and a 2-cocycle $\alpha\in Z^{2}(G,A)$. Recall that an $...

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389 views

### The number of maximal subgroups up to isomorphism

Every maximal subgroup of infinite index of a free non-cyclic group $F_k$ is free of countable rank. Thus even though the set of maximal subgroups of $F_k$ is uncountable, there are only countably ...

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80 views

### Finite quotients of amalgamated products with virtually nilpotent factors

Consider the amalgamated product $A\ast_C B$ of groups such that $A\neq C\neq B$ and both factors $A$, $B$ are finitely generated virtually nilpotent.
Does $A\ast_C B$ always have a subgroup of some ...

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110 views

### Quotient of Coxeter complex in terms of double cosets?

In Victor Reiner's Quotients of Coxeter Complexes and $P$-Partitions, we have the below definition for the quotient complex of a Coxeter complex by a finite subgroup of the Coxeter group. I think ...

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267 views

### Groups whose finite index subgroups of fixed index are isomorphic

I am interested in finitely generated groups $G$ that are residually finite and have the following property: For each $d \geq 1$, $G$ has subgroups of finite index $d$, and all such subgroups are ...

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570 views

### Geometric or topological results from group theory

Do you know interesting examples of purely geometric or topological results which can be proved using group theory? To make precise what I have in mind, let us consider the two following examples:
...

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260 views

### Finite groups with lots of conjugacy classes, but only small abelian normal subgroups?

Denote the commuting probability (the probability that two randomly chosen elements commute) of a finite group $G$ by $\operatorname{cp}(G)$. By a result of Gustafson [2], $\operatorname{cp}(G)=\...

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314 views

### When are growth series rational?

For a finitely generated group $G$ with generating set $S$, one can define the word metric. Using this, we are able to define the sequence $\sigma(n)$ by
$ \sigma(n) = |\mathcal{S}(e,n)| $, which is ...

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176 views

### A spectral graph theory problem

Let $S$ be a zero-free subset of the group ${\bf Z}_2^n$ and $\Gamma={\rm Cay}({\bf Z}_2^n,S)$ be a bipartite Cayley graph. For some choices of $S$, the graph $\Gamma$ has $4$ distinct eigenvalues, ...

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83 views

### Generating-bijective groups

We may say that two finitely generated groups $G$ and $H$ are generating-bijective when there exist homomorphisms $\phi:G\rightarrow H$ and $\psi:H\rightarrow G$ such that, for each ordered generating ...

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183 views

### When does a CAT(0) group contain a rank one isometry

Let $G$ be a CAT(0) group which acts on the CAT(0) space $X$ properly and cocompactly via isometry. Let $g \in G$ be a hyperbolic isometry of $X$. Then $g$ is called $\textbf{rank one}$ if no axis of $...

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147 views

### For $P$ $\mathbb{Z}G$-projective, $\mathbb{Q}\otimes P$ is $\mathbb{Q}G$-free

I'm looking for a proof of a theorem of Swan [1, Theorem 3]:
If $G$ is a finite group and $P$ a finitely generated projective $\mathbb{Z}G$-module, then $\mathbb{Q}\otimes_\mathbb{Z}P$ is a free $...

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134 views

### Number of eigenvalues of a Cayley graph

Let $G=Z_2^n$ and $S\subset G$. Is there any relation for number of distinct eigenvalues of $\Gamma=Cayley(G,S)$ graph depending on $n$ and $|S|$, or at least diameter of $\Gamma$? If you have any ...

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160 views

### Rank of a locally free $\mathbb Z[G]$-module

This is a basic question. Let $G$ be a finite group, $M$ a finitely generated $\mathbb Z[G]$-module so that the $\mathbb Z_p[G]$-module $M_p$ is free for all prime numbers $p$, i.e. is locally free.
...

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542 views

### Does $E_8$ know $Spin(7)$?

One way to define the compact group $Spin(7)$ is as the stabilizer of a certain 4-form on Euclidean $\mathbb R^8$ (see e.g. this MO question). This 4-form can be defined in various ways. For example,...

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224 views

### A finiteness property for semi-simple algebraic groups

Let $G$ be a semi-simple algebraic group over a field $K$, I am considering a question about whether there exists a finite set of semi-simple $K$-subgroups, say $H_1,...,H_r$, such that for any semi-...

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230 views

### Weil's Haar measure construction from below

Weil's construction of a Haar measure on a locally compact group rests on approximating a function from above by sums of translates of another function.
I would need to know something similar for an ...

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119 views

### Constructing the largest finite group with a fixed number of conjugacy classes

It is known that there are finitely many finite groups with a given number of conjugacy classes. How can one construct (or get a character table for) the groups $G$ that realize the maximum possible ...

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### Is $n+\frac {1}{2}$ in Kendall-Mann numbers and quantum harmonic oscillator related?

It is known that quantum harmonic oscillator is connected to the symmetric group of infinite order which is isomorphic to the permutation group. According to Cayley's theorem any finite group is ...

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### Hemi-Semi Direct Product

In the category of racks (similarly quandles), instead of well-known semi direct product, we have hemi-semi direct product construction as seen on Wagemann & Crans.
As far as I know, semi direct ...

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91 views

### Extension property for unipotent linear groups over rings

This is my first question, so my apologies if it is too simple/poorly motivated.
During the course of some recent research I came across a particular variant of the following problem.
Let $G$ ...

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### Characteristically simple locally compact abelian groups

Say a topological group $G$ is topologically characteristically simple if there does not exist a closed subgroup $1 < K < G$ such that $K$ is invariant under all automorphisms of $G$ (here `...

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140 views

### Are braid groups conjugacy separable?

I would like to re-ask a question that was raised in the comments here:
Normal subgroups of braid groups
Namely, a group $G$ is called conjugacy separable, if it holds that for every two elements $x,...

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90 views

### Finitely presented amenable LERF group which is not virtually solvable

Is there a group $G$ with the following properties?
Finitely presented
Amenable
Not virtually solvable
LERF (that is, every finitely generated subgroup is closed in the profinite topology on $G$).
...

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478 views

### A question about (unicity of certain cycles in a Cayley graph of a) symmetric group

Let $S=\{(1,2),(1,2,3,\ldots,n),(1,2,3,\ldots,n)^{-1}=(1,n\ldots,2)\}$ be a subset of the symmetric group $S_n$. We know that $(1,2,\ldots,n)(1,2)=(2,3,\ldots,n)$, and thus $$[(1,2,\ldots,n)(1,2)]^{n-...

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304 views

### Lower central series quotients in terms of (co)homology

Let $G$ be a group. It is well-known that $H_1(G,\mathbb{Z})=G/[G,G]$. Also (at least up to torsion) $[G,G]/[G,[G,G]]=\Lambda^2H^1(G,\mathbb{Z})/H_2(G,\mathbb{Z})$ as explained, for example, in this ...

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160 views

### Properties of a special finitely presented groups

Recently, when I was working with Cayley graphs, I faced up with a special group. The original group is as follows:
$$G:=<a,b,c|ab=ba,a^{10}=cbc^{-1}>.$$
We can show that this group can be ...

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67 views

### Limits of quotient groups in the space of marked groups

In the space of marked groups with $m$ generators, suppose that a sequence $(G_i, S_i)$ converges to $(G, S)$. For any $i$, let $K_i$ be a normal subgroup of $G_i$ and assume that $\bar{S}_i$ is the ...