Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
7,942
questions
3
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A question about Gromov's proof of a "more effective version of the main theorem"
In the paper "Groups of polynomial growth and expanding maps" Gromov proves the following "effective version of the main theorem"
For any positive integers $d$ and $k$, there ...
2
votes
0
answers
89
views
Are the alternating pentultimate and the $2$-alternating pentultimate isomorphic?
Consider a physical puzzle which is in the shape of a dodecahedron where the pieces that move are the face centers and you can slice it through a plane parallel to two opposite faces going through the ...
6
votes
1
answer
369
views
Do acyclic amenable groups exist?
Is there an example of a nontrivial discrete amenable group with vanishing integral homology?
To put the question in contrapositive. Given arbitrary acyclic group $Q$, is there some reason for the ...
4
votes
0
answers
191
views
Polynomials of growth for finite Heisenberg groups
Take a standard finite Heisenberg group with two standard generators and let's consider its growth polynomial - the polynomial which coefficients are equal to the sphere sizes.
For example for $H_3(Z/...
6
votes
2
answers
404
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Embedding $\mathrm{SL}_n(3)$ into $\mathrm{SL}_n(p)$
$\DeclareMathOperator\SL{SL}$Let $p$ be an odd prime. It is easy to show that $\SL_2(3)$ can be embedded into $\SL_2(p)$.
Now, let $n$ be an integer larger than $2$.
Question: In which circumstances, $...
3
votes
2
answers
213
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Factorizations of an $n$-cycle in $S_n$ into a product $xy$ where $|x| = 2, |y| = 3$
Let $S_n$ be the symmetric group on $n$ letters. When (and how) can an $n$-cycle in $S_n$ be factored into a product $xy$, where $x,y$ have orders 2,3 respectively?
More precisely, I'd like to ...
4
votes
0
answers
396
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Problem 1.8 from Kirby's list
Context
I looked through a book called "Problems in Low-Dimensional Topology", where Rob Kirby lists a set of problems. He provides a list of problems, states their conjectures, and ...
0
votes
0
answers
35
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Growth of cocycles in higher degrees
Let $G$ be a group with finite symmetric generating set $S$ and
let $\pi:G\rightarrow\mathcal{U}(\mathcal{H})$ be a unitary representation
of $G$ on a Hilbert space $\mathcal{H}$. A 1-cocycle with ...
9
votes
3
answers
321
views
$G$-module structure of the relation module for a presentation of a finite group $G$
Let $F_n$ be a free group of rank $n\ge 2$, and $F_n\rightarrow G$ a surjection with $G$ finite. Let $R$ be the kernel. From this, we get an action of $G$ on the abelianization $R/R'$ (a free abelian ...
7
votes
0
answers
365
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Subgroups of ${\rm GL}_{2}(\mathbb{Z}/2^{n} \mathbb{Z})$
Assume that $n \geq 3$. Is there a subgroup $H \leq {\rm GL}_{2}(\mathbb{Z}/2^{n} \mathbb{Z})$ whose order is a power of $2$ so that
$\bullet$ $\det : H \to (\mathbb{Z}/2^{n} \mathbb{Z})^{\times}$ is ...
0
votes
1
answer
171
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Reflections on subspaces of $\text{codim} > 1$
Let $V$ be a real finite-dimensional vector space with inner product $\langle \cdot , \cdot \rangle$.
Let $x,y \in V$ be linearly independent. I was wondering how a reflection $s_{x,y}$ through the $\...
0
votes
0
answers
148
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Residual finiteness of semidirect product $\mathbb{Z}^2\ltimes \mathbb{Z}[1/10]$ of abelian groups
Let $\mathbb{Z}[1/10]$ be an abelian group by addition. Let $\mathbb{Z}^2$ act on it by automorphisms by $x\mapsto 2x$ and $x\mapsto 5x$. Is the corresponding semidirect product $\mathbb{Z}^2\ltimes \...
5
votes
2
answers
202
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What are the Schur indices of irreducible representations of $\operatorname{SL}(2,p)$?
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$What are the Schur indices of the irreps of $\SL(2,p)$? ($p$ an odd prime.)
Presumably this is in a book somewhere? Section 6 of the paper &...
1
vote
0
answers
157
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Applications of Artin's theorem on induced representations
Let $G$ be a finite group and let $R(G)$ be the (complex) representation ring of $G$. As stated in Serre's book on representation theory, Artin's theorem says the following:
Theorem: Let $X$ be a ...
0
votes
1
answer
188
views
Fixed points free automorphisms of Teichmüller spaces
Let $\mathcal{T}_{g,n}$ be the Teichmüller space of a compact oriented surface of genus $g$ with $n$ marked points. Assume that $N:=3g-3+n>0$. Viewing $\mathcal{T}_{g,n}$ as a bounded domain in $\...
6
votes
1
answer
412
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Relations between relations in the positive braid monoid
The positive braid monoid on $n$ strands is the monoid with generators $s_1$, $s_2$, ..., $s_{n-1}$ and relations
$$s_i s_{i+1} s_i = s_{i+1} s_i s_{i+1} \qquad s_i s_j = s_j s_i \text{for}\ |i-j| \...
4
votes
3
answers
554
views
Regular orbits for automorphisms of finite simple groups
Let $G$ be a finite group and $f$ be an automorphism of $G$. We say that $f$ has a regular orbit if there exists $x\in G$ such that $|x^f|=|f|$. If $G$ is abelian it is known that every automorphism ...
4
votes
0
answers
197
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Infinite groups with 2 automorphism orbits
A group $G$ is called a $k$-orbit group if its automorphism group ${\rm Aut}(G)$ acting naturally on $G$ has precisely $k$ orbits. The only finite 2-orbit groups are the elementary abelian groups $(...
14
votes
0
answers
475
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Is the monster group maximal in SO(196883)?
$\DeclareMathOperator\SO{SO}$The smallest degree of a nontrivial complex representation of the monster group $ M $ is $ 196883 $. This irrep has Schur indicator $ 1 $, so the image must lie in the ...
7
votes
1
answer
637
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What are double groups mathematically?
In physics and chemistry, there is the concept of double groups. These are double covers of the usual point groups, obtained by "adding an element $R$, which represents rotation by $2\pi$" ...
0
votes
1
answer
169
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Are all "almost projective" groups free?
Let us say that a group $H$ is almost projective if, given any group epimorphism $f\colon G\to H$, there is an embedding $i\colon H\to G$.
Does it follow that $H$ is free? If not, is there a ...
9
votes
1
answer
508
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Shortest almost trivial element of free group [duplicate]
Let $F_n$ be the free group with $n$ generators $\gamma_1,\dots,\gamma_n$.
Consider the homomorphisms $h_i\colon F_n\to F_{n-1}$ defined by adding the relation $\gamma_i=1$ in $F_n$.
What is the ...
9
votes
1
answer
374
views
Is every automorphism of $\mathrm{Aut}^+(F_2)$ induced by conjugation inside $\mathrm{Aut}(F_2)$?
$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Inn{Inn}\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}$Let $F_2$ be a free group of rank 2. There is a surjection $\Aut(F_2)\rightarrow \GL(2,...
2
votes
1
answer
225
views
An interior cone condition for Teichmuller spaces
Let $\mathcal{T}_{g,m}$ be the Teichmuller space of a compact oriented surface of genus $g$ with $m$ marked points. Consider it as a bounded domain in a complex space $\mathbb{C}^N$. Let $\xi$ be a ...
3
votes
1
answer
228
views
Nonisomorphic central products on the same pair of groups?
A central product of two groups $G$ and $H$ is determined as follows. The groups $G$ and $H$ have respective central subgroups $A$ and $B$ which are isomorphic, let $\delta:A\rightarrow B$ be such ...
2
votes
0
answers
91
views
What is the complexity / name of word search problem in linear groups?
This is a question about a search problem associated with user6976's question. Suppose we are given a finite set of elements $S \subset \mathrm{GL}_n(\mathbb{Q})$ containing inverses of all its ...
3
votes
1
answer
268
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Distinct characters with the same character values, outer automorphisms and Galois conjugation
Given an (irreducible complex) character of a finite group the following three construction all yield another irreducible character of the same degree:
multiplying by a degree 1 character
applying an ...
1
vote
0
answers
63
views
Commutative Schur ring over non-abelian group
Is there a commutative (or even symmetric) Schur ring $S\subset\mathbb{C}G$ over a non-abelian group $G$, which is not isomorphic (preserving both the products) to a Schur ring $S'\subset\mathbb{C}G'$ ...
4
votes
0
answers
135
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Vanishing of $\ell^2$-Betti numbers of $\mathrm{GL}(n,\mathbb{Z})$ for $n\geq 3$
$\DeclareMathOperator\GL{GL}$In a paper I read the following claim:
By the work of Borel the $\ell^2$-Betti numbers of the cocompact lattices of $\GL(n,\mathbb{R})$ are known to all vanish when $n ≥ 3$...
1
vote
1
answer
336
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Distribution of 2-groups
In the family of finite groups of order less than $2000$, there are about 99% of order $1024$, so I have a question about $2$-groups:
Let $f(n)$ be the number of non-isomorphic finite groups of order $...
13
votes
4
answers
680
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Prove these are not surface groups
For $g,n \geq 1$, let $\Gamma_{g,n}$ be the group with the following presentation:
$$\langle \text{$a_1,b_1,\ldots,a_g,b_g$ $|$ $[a_1,b_1]^n [a_2,b_2] \cdots [a_g,b_g]=1$} \rangle.$$
For $n = 1$, ...
6
votes
0
answers
115
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Sylow subgroups of the restricted Burnside group $\mathrm{RB}(d,n)$?
$\DeclareMathOperator\RB{RB}$What is known about the Sylow subgroups of the restricted Burnside groups $\RB(d,n)$ ?
I am looking for a reference.
In fact my question is slightly more general. Recall ...
6
votes
1
answer
216
views
Classification of non-abelian simple groups with cyclic T.I. Sylow p -subgroup
Let $G$ be a finite group and $H$ a subgroup of $G$. We say that $H$ is a trivial intersection (for short T.I.) subgroup of $G$
if $H\cap H^x=1$ for each $x\in G-N_G(H)$.
I read the next result in the ...
1
vote
1
answer
72
views
Compact subgroups of a linear group over non-Archimedean local field
$\DeclareMathOperator\GL{GL}$Let $\mathbb{F}$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers.
Is it true that any compact subgroup of $\GL_n(\mathbb{F})$ is conjugated to ...
4
votes
0
answers
189
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$\mathrm{Out}(G)\times\mathrm{ Out}(H)$ is a finite index subgroup of $\mathrm{Out}(G\times H)$
When $G$ and $H$ are indecomposable centerlesss groups it is known that $\mathrm{Aut}(G)\times \mathrm{Aut}(H)\cong\mathrm{Aut}(G\times H)$ if $G\ncong H$ and $\mathrm{Aut}(G)\times \mathrm{Aut}(H)$ ...
1
vote
0
answers
166
views
Are these sequences, associated to integer partitions, always log-concave?
Let $\mathrm{CO}(m)$ be the set of all compositions of the
positive integer $m$. By a composition of $m$, I mean a finite
sequence $(m_1,\ldots,m_k)$ of positive integers with sum $\sum
m_i=m$.
...
4
votes
1
answer
360
views
Finite groups with bounded centralizers
Let $G$ be a finite group. For each $x\in G$, the centralizer $\mathbf{C}_G(x)$ must contain $\langle x\rangle$.
QUESTION: What are some interesting results of the following form:
Given some bound on $...
2
votes
0
answers
124
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Automorphisms of (nilpotent) groups : torsion cokernel on the abelianisation implies torsion cokernel on the center?
$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\End{End}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Z{Z}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\Span{Span}\DeclareMathOperator\ker{ker}\...
5
votes
1
answer
245
views
Word length in the surface groups
I want to know if there are some results about the title of this question.
Let $G$ be an orientable closed surface group with genus $n$ greater than 1. We know it has a canonical presentation.
$$G=\...
2
votes
0
answers
79
views
Splitting of $\mathbb{Z}/p\to E\to (\mathbb{Z}/p)^n$ in cohomological terms
Let $d>1$ be an odd integer. Given a simplicial set $X$ and $[\gamma]\in H^2(X,\mathbb{Z}/d)$, there exists a fibration $N\mathbb{Z}/d\to E\to X$, with $$E= X_\gamma:=N\mathbb{Z}/d\times_{\gamma} X....
5
votes
0
answers
192
views
Virtual fibring of $\mathrm{Out}(F_2\times F_2)$
A finitely generated group $G$ is said to virtually fibre if there is a finite index subgroup $H\leq G$ and a non-trivial map $\varphi:H\to\mathbb{Z}$ with $\ker(\varphi)$ finitely generated.
I want ...
7
votes
1
answer
457
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Computing homology groups with GAP
I’m studying the homology groups of arithmetic groups such as $SL(5,\mathbb{Z})$. I saw in the answer to this post that we can use GAP to compute some of the homology groups for $SL(3,\mathbb{Z})$. Is ...
3
votes
1
answer
93
views
Hamiltonian paths in 2-generated (Cayley) circulant digraphs. A counterexample?
Circulant digraphs are Cayley digraphs of cyclic groups. My question refers to hamiltonian paths (not to hamiltonian cycles) in 2-generated circulant digraphs (not graphs).
There is a theorem by ...
2
votes
0
answers
192
views
Existence of a finitely presented simple group satisfying certain properties
Is there a finitely presented simple group with exactly 8 conjugacy classes of finite subgroups, which have the following respective isomorphism types?
$C_1$
$C_2$
$C_3$
$C_2^2$
$C_6$
$S_3$
$A_4$
$...
2
votes
0
answers
136
views
A possible generalization of "homotopy" to study group actions of various kinds
This is a naive question about abstract homotopy theory by someone who knows nothing about it, except that it involves some generalization of the notion of "homotopy".
If we think of $O(n)$ ...
0
votes
0
answers
71
views
Lifting of algebraic fibrations via a subnormal series
Consider a finitely generated group $G$ and a subnormal series of $G$:
$$1=G_0\trianglelefteq G_1\trianglelefteq\cdots\trianglelefteq G_{n-1}\trianglelefteq G_n=G$$
Now, suppose that $G_1$ fibres, i.e....
1
vote
0
answers
81
views
$L^p$-compression of metabelian groups
Question: Is there a metabelian group, so that for some $\epsilon >0$ and all $p \in [1, \infty[$ the [equivariant] compression exponent in [any] $L^p$-space is bounded by $1-\epsilon$ (bound does ...
6
votes
1
answer
343
views
All surjections onto trivial irrep split equivalent to being reductive
$\DeclareMathOperator\Hom{Hom}$Let $ G $ be linear algebraic group over a field $ k $. Is it true that every short exact sequence of algebraic $ G $-representations
$$
0 \to W \to V \to k \to 0
$$
...
5
votes
1
answer
331
views
Lattice generated by parabolics
Let $G$ be a semisimple Lie group of split-rank one and let $\Gamma$ be a non-cocompact lattice which is torsion-free.
For the group $G=\mathrm{SL}_2(\mathbb{R})$ it then follows that $\Gamma$ is ...
7
votes
1
answer
254
views
Relation between Floyd and Gromov boundaries of hyperbolic groups
Let $G$ be a hyperbolic group. Consider the Floyd boundary as defined in https://www.unige.ch/math/folks/karlsson/free.pdf by Karlsson. For a Floyd function $f$, we denote the Floyd boundary of $G$ by ...