All Questions
Tagged with reference-request gr.group-theory
700 questions
16
votes
3
answers
2k
views
Your favorite papers on geometric group theory
I would like to improve my "depth of understanding" in geometric group theory. So I am interested in short and accessible papers on subjects related to this field but which are not always ...
16
votes
2
answers
818
views
Decomposing $(\mathbb C^n)^{\otimes m}$ as a representation of $S_n\times S_m$
$V=\mathbb C^n$ is a $\mathbb CS_n$-module, where $S_n$ is the symmetric group of degree $n$, via the representation sending a permutation to the corresponding permutation matrix. The tensor power $V^...
- 38.6k
10
votes
1
answer
381
views
About the paper by Buekenhout, Delandtsheer, Doyen, Kleidman, Liebeck and Saxl
The paper by Buekenhout, Delandtsheer, Doyen, Kleidman, Liebeck and Saxl called Linear spaces with flag transitive automorphism groups (Geom. Dedicata) from 1990 annonces a very powerful ...
- 2,287
3
votes
2
answers
279
views
Searching for theorems characterizing when $O_p(G)$ is trivial / non-trivial
Let $G$ be a finite group. Let $p$ be a prime.
Let $O_p(G)$ be the $p$-core of $G$.
Are there any theorems known saying something like
$O_p(G)$ is trivial, if and only if ... and
$O_p(G)$ is non-...
- 237
10
votes
2
answers
815
views
Paper by I. N. Sanov, Solution of the Burnside problem for exponent 4
I have searched extensively online and for copies of printed journals containing the paper which details Sanov's solution to the Burnside Problem for exponent 4, which is widely cited in many papers ...
- 201
5
votes
0
answers
172
views
Finitely generated nilpotent groups with hyperbolic automorphisms
$\DeclareMathOperator\Out{Out}\DeclareMathOperator\GL{GL}$
Let $G$ be a finitely generated nilpotent group.
We call an automorphism of $G$ hyperbolic if the induced automorphism of the free part of ...
- 8,529
9
votes
2
answers
772
views
Characters of orthogonal groups as symmetric functions
This question was asked on MSE some time ago, here, but got no attention.
The Schur functions are characters of irreps of the unitary group, $s_\lambda(U)=Tr(R_\lambda(U))$. They are symmetric ...
- 2,552
9
votes
0
answers
456
views
Hopficity of Baumslag-Solitar groups
I am struggling to find the exact source (with proofs) of the following ''well-known'' statement:
the Baumslag-Solitar group $BS(m,n)=\langle a,t \mid ta^m t^{-1}=a^n\rangle$ is Hopfian if and only if ...
- 3,181
6
votes
1
answer
693
views
Finite subgroups of $GL(2,K)$ with $K\neq\mathbb{C}$
It is well known that the finite subgroups of $SL(2,\mathbb{C})$ up to conjugacy are the binary polyhedral groups (or Klein groups). There are two infinite families (cyclic groups and binary dihedral ...
- 411
27
votes
1
answer
3k
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An anecdote by R. Schmidt
Did anybody here ever read those lines by R. Schmidt (?) where he talked about the terseness of articles in group theory in the days prior to the conclusion of the classification of the finite simple ...
1
vote
2
answers
287
views
Faithfully flat modules over a group algebra
Suppose we have the following data:
1) A group ring $\mathbb{Z}[G]$, where $G$ is a torsion free group.
2) $M_{\bullet}$ a bounded (above and below) chain complex of $\mathbb{Z}[G]$-modules such ...
- 71
0
votes
0
answers
169
views
A group acts on a groupoid
Let $G$ be a group. Let $(\Pi,\circ)$ be a groupoid. Suppose I have a $G$-action on every morphism space $\Pi(p,q)$, denoted by $G\times \Pi(p,q)\to \Pi(p,q)$, $(g, \sigma)\mapsto g\cdot \sigma$. (For ...
- 2,789
3
votes
0
answers
239
views
Groups with "just not" a property
There seems to be a standard trick in group theory which is to show that a group has a quotient group which "just not" has some property.
To make things clear:
let $\mathcal{P}$ be a group ...
- 4,432
5
votes
5
answers
873
views
Green polynomials
Is there any software for calculating Green polynomials (of type A)? Or, at least, where can I find tables of Green polynomials? Also, I would be interested in some formulas for Green polynomials in ...
- 1,590
10
votes
4
answers
2k
views
Conjugation Quandles and... "Quandle-Groups"? From quandles to Groups
This question is already asked MathSE
A quandle $(Q,*,/ )$ is a idempotent right-distributive and right invertible structure.
1) $a*a=a$
2) $(a*b)*c=(a*c)*(b*c)$
3) $(a*b) /b=(a/b)*b=a$
...
- 233
2
votes
1
answer
129
views
Source for: a permutation group is multiplicity-free if and only if its 2-orbits define an association scheme
I have recently proven the following (at least, so I believe):
Theorem. Given a permutation group $\Sigma\subseteq\mathrm{Sym}(\Omega)$ on the set $\Omega:=\{1,...,n\}$, the following are equivalent:
...
- 13.6k
1
vote
0
answers
340
views
Random walk on non-abelian free group
Let $F_2$ be the free non-abelian group with generators $a, b\in F_2$.
Has the "random walk" where we start with the identity and then multiply it by $a$ or $b$ or $a^{-1}$ or $b^{-1}$ ...
- 11
24
votes
3
answers
3k
views
Non-abelian Grothendieck group
By general nonsense the forgetful functor from groups to monoids has a left adjoint. It maps a monoid $(X,\cdot,1)$ to the free group on $\{\underline{x} : x \in X\}$ modulo the relations $\underline{...
- 63.1k
5
votes
1
answer
396
views
Finite simple groups with three conjugacy classes of maximal local subgroups
$\DeclareMathOperator\PSL{PSL}$In [1] it was proved that
A finite nonsolvable group $G$ has three conjugacy classes of maximal subgroups if and only if $G/\Phi(G)$ is isomorphic to $\PSL(2,7)$ or $\...
- 151
5
votes
1
answer
279
views
Permanent of a Kronecker product of matrices
It is well known that $\det(A \otimes B) = \det(A)^m \det(B)^n$ when $A$ and $B$ are square matrices of size $n$ and $m$ where $\otimes$ denotes the Kronecker product.
Question: Is there a similar ...
- 26.5k
11
votes
1
answer
688
views
Unitary representations of finite groups over finite fields
I would like to learn the basic theory of unitary representations of finite groups over finite fields.
Here, the unitary group $\operatorname{GU}(n,\mathbb{F}_{q^2})$ consists of all invertible ...
- 111
9
votes
4
answers
1k
views
Symmetries of probability distributions
When talking about a single random variable, knowing only its distribution, the construction of a probability space is quite easy. Namely, let $(X,\mathscr A)$ be a measurable space and let $\mathsf Q$...
- 1,655
9
votes
1
answer
2k
views
Finite groups in which all proper subgroups are cyclic
Is there any classification of finite group in which all proper subgroups are cyclic?
Would you please tell me a reference?
- 91
4
votes
1
answer
436
views
Connectedness of the stabilizer in a semisimple group of a semisimple element in the Lie algebra: a reference request
Let $G$ be a (connected) semisimple algebraic group over an algebraically closed field $k$ of characteristic 0.
We consider the adjoint representation
$$ {\rm Ad}\colon G\to {\rm GL}({\mathfrak g}),$$
...
- 14.2k
0
votes
0
answers
250
views
Has this theorem on cancellative monoid actions been discovered and published?
Does a statement equivalent to Theorem 3 below appear in the literature? If it does, what is the earliest published reference?
Theorem 1. Let $W$ be a non-trivial cancellative invertible-free [1] ...
4
votes
1
answer
239
views
Latest progress on Tarski numbers
Two questions, the first: What is the smallest non negative integer that we do not know yet is the Tarski number of a group?
The second question is the same as in the title: What is the latest ...
- 2,118
5
votes
1
answer
152
views
How to find a finite splitting field $K$ for $G$ such that every indecomposable $KG$-module is absolutely indecomposable
Let $G$ be a finite group and let $k$ be a finite field with char$(k)=p$ such that $p\mid |G|$.
If $k$ is a splitting field for $G$, then, no matter which splitting field we take, after extending ...
- 1,755
3
votes
1
answer
276
views
For $G$ an adjoint Chevalley group, are all of $G(\mathbb Z)$'s finite-index subgroups congruence subgroups?
Let $G$ be an adjoint Chevalley group. Are all of $G(\mathbb Z)$'s finite-index subgroups congruence subgroups?
I read a theorem that states: When $G$ is the universal Chevalley group and it's not of ...
- 332
3
votes
0
answers
153
views
Metropolis-Hastings sampling as a group action
Suppose that you have a topological space $\Omega \subset \mathbb R^n$ accompanied a measure $\mu$ and you're running an iterative sampling algorithm like Metropolis-Hastings. To sample you choose a ...
12
votes
2
answers
341
views
Which $K$-groups $K(C^*_r(G))$ are computed?
We have the Pimsner-Voiculescu exact sequences and the Baum-Connes map
for possible computation of the $K$-theory of the reduced group $C^*$-algebra $C^*_r(G)$ for a topological, locally compact, ...
- 685
4
votes
0
answers
177
views
Ping pong with parabolic isometries on Gromov hyperbolic spaces
For a group $G$ with a non-elementary general type action by isometries on a Gromov hyperbolic geodesic space $(X,d)$, it is well known that you can construct free subgroups of $G$ via the ping pong ...
- 41
15
votes
1
answer
1k
views
quasi-homomorphisms of groups
Suppose that $G$ is a group and $d$ is a left-invaraint metric on $G$, e.g., the word metric (provided that $G$ is finitely-generated) or distance function determined by a left-invariant Riemannian ...
- 31.2k
1
vote
0
answers
92
views
Group structure extension
Let $G$ be a finite group and $X$ a finite $G$-set. Let $H$ be the set-theoretical cartesian product of $G$ and $X$.
Is there an homological theory controlling all possible group structure on $H$ (...
- 2,384
6
votes
1
answer
235
views
Introductory text on amenability
I am looking for a book that covers amenability rigorously.
Preferably a book aimed at beginners.
- 5,450
4
votes
1
answer
256
views
On $(2,3)$-generation of finite simple classical groups
A group $G$ is called $(a,b)$-generated if $G=\langle x,y\rangle$ for some $x,y\in G$ with $|x|=a$ and $|y|=b$.
I know some of the histories on this problem. For example, in this early paper in 1996 ...
- 379
1
vote
0
answers
213
views
Is there any research on the action of a subgroup on the whole finite group by conjugation?
I want to know whether there are any research on the orbits of the action of a subgroup by conjugation on the whole group, when the group is finite. (Especially whole symmetric group.)
I'm especially ...
- 1,013
6
votes
0
answers
245
views
A group action on another group action quotient: how to best describe the resulting structure and does it have a name?
Suppose I have an action $\alpha:G\times X\to X$ of a group $G$ on a set $X$ and, on top of that, an action $\beta:H\times(X/G)\to(X/G)$ of another group $H$ on the set of $G$-orbits.
Is there a nice ...
- 18.4k
16
votes
3
answers
1k
views
What are some interesting examples of non-classical dynamical systems? (Group action other than $\mathbb{Z}$ or $\mathbb{R}$ )
By classical dynamical system, I mean a measure space together with a measurable action of the integers or the reals. Of course, this action is often interpreted as evolution with respect to discrete ...
1
vote
0
answers
96
views
Embedding a family of groups into a certain $2$-generated group (construction by Olshanskii)
While reading "Chain conditions, elementary amenable groups, and descriptive set theory" by Phillip Wesolek and Jay William I stumbled upon the following statement in the proof of ...
- 309
3
votes
1
answer
302
views
Deciding isometry of unimodular lattices by Gram matrices
Say I have two unimodular lattices $A$ and $B$, represented by their Gram matrices.
Question: Is there an algorithm to decide whether $A$ and $B$ are isometric, i.e. whether there exists a matrix $S \...
- 9,501
3
votes
2
answers
676
views
Primitive action of wreath product
I say in advance that I am really new to Group Theory, so if my question is trivial I apologize in advance.
Let $A$ and $H$ be groups and $\Omega$ be a $H$-set. In this set-up, we can define the ...
- 1,252
9
votes
2
answers
571
views
Algorithm for group cohomology
Let $G$ be a finite group, and let $0\to M_1\xrightarrow{\iota} M_2\xrightarrow{\pi} M_3\to 0$ be a short exact sequence of $G$-modules (finitely generated over $\mathbb Z$, not necessarily free). I ...
- 161
9
votes
2
answers
2k
views
alternating and symmetric powers of the standard representation of the symmetric group
Let $n \geq 7$ and $V = \mathbb{C}^n$ be the standard representation for $S_{n+1}$, the symmetric group of cardinal $(n+1)!$
Let $k$ be an integer such that $2 \leq k \leq n$. Is it true or false ...
- 7,300
3
votes
0
answers
115
views
Reference for the Netto's theorem on the permutation groups which was mentioned in the paper of Frobenius
I'm trying to read 'Uber die Charaktere der mehrfach transitiven Gruppen' written by Frobenius.
There he mentioned some theorems of Netto.
I'm depending on the Google translator. and the translation ...
- 1,013
0
votes
0
answers
167
views
Is there a theory of "partial" group actions?
I am looking for references that may formalize the following idea: let $R = k[X]$ be the coordinate ring for a generic $n \times m$ matrix $M$. It is well known that the ideal of $r \times r$ minors ...
- 553
11
votes
4
answers
1k
views
Examples of acylindrical 3-manifolds
Let $C$ be the compact cylinder $S^1\times [0,1]$. A 3-manifold $M$ with incompressible boundary is called acylindrical if every map $(C,\partial C)\to (M,\partial M)$ that sends the components of $\...
- 25k
13
votes
3
answers
2k
views
Weil's book L'intégration dans les groupes topologiques et ses applications
The book L'intégration dans les groupes topologiques et ses applications published by André Weil in 1940 is regarded as one of the classical references for harmonic analysis on topological groups.
...
user92170
17
votes
5
answers
3k
views
Reference for this theorem in representation theory?
Let $G$ be a finite group and $\chi$ be an irreducible character of
$G$ (characteristic zero algebraically closed base field). If $H$ is
the kernel of $\chi$ then the irreducible representations of $G/...
- 887
8
votes
0
answers
125
views
The conjugacy problem for two-relator groups
Is the conjugacy problem for two-relator groups known to be undecidable?
The word problem for two-relator groups is a famous open problem (appearing e.g. as Question 9.29 in the Kourovka notebook), ...
8
votes
0
answers
346
views
A generalization of Feit–Thompson conjecture, for square-free integers
I asked the following question with my account that I have for these sites Mathematics Stack Exchange and MathOverflow. The bounty that I offered in MSE expired without answers. The post that I refer ...
