All Questions
Tagged with reference-request gr.group-theory
700 questions
7
votes
1
answer
697
views
Growth of Thompson's group $F$
EDIT(August 2013): I accepted Mark's answer as being the state of art- there are two relevant references, one in the answer and one in the comments. The minimal growth rate of $F$ remains unknown with ...
- 992
9
votes
1
answer
337
views
amenable + without $BS(m,n)$+finite $K(G,1)$implies virtually cyclic?
I heard from someone that the following problem is an open question.
(Open Problem 1)For a countable discrete group $G$, suppose it does not contain any Baumslag-Solitar subgroups $BS(m,n):=\langle x,...
- 1,528
10
votes
2
answers
3k
views
Doubly-transitive groups
I want to know what all doubly-transitive groups look like. Do you know some good reference where I can read about it?
- 2,179
10
votes
3
answers
1k
views
Subgroups of GL_2 over a finite field
I've come across the phrase "by the classification of subgroups of $GL_2(F_q)$" in multiple papers, but never with a reference. Here $F_q$ is a finite field of size $q$. Does anyone know a good ...
- 101
2
votes
1
answer
975
views
Explicit examples of Dehn presentations of hyperbolic groups
It is well known fact that a (f.g.) group is hyperbolic if and only if it admits a (finite) Dehn presentation.
My question concerns something I'm struggling with since the first time I read the proof ...
- 721
8
votes
1
answer
229
views
Embedding abelian cancellative Hausdorff topological semigroups into abelian Hausdorff topological groups
An abelian cancellative semigroup embeds (via a semigroup monomorphism) into an abelian group. What about an abelian cancellative Hausdorff topological semigroup that does not embed (via a ...
- 10.5k
2
votes
2
answers
381
views
Speed and absence of non-constant bounded harmonic functions
For a (symmetric) random walks on countable groups generated by $\mu$, there is a "brute-force computation" argument of Avez (1974) that shows that if the entropy $h_\mu$ is trivial then there are no ...
- 4,432
3
votes
0
answers
269
views
Reference for the rank of the BN-pair of the finite simple groups of Lie type and not Chevalley
The rank of the BN-pair of a Chevalley group is the number of simple roots of its Lie algebra, which is the index of the name of its Dynkin diagram ($A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4,G_2 $).
...
6
votes
2
answers
486
views
Centralizers of reflections in special subgroups of Coxeter groups
Let us consider a (not necessarily finite) Coxeter group $W$ generated by a finite set of involutions $S=\{s_1,...,s_n\}$ subject (as usual) to the relations $(s_is_j)^{m_{i,j}}$ with $m_{i,j}=m_{j,i}$...
13
votes
1
answer
1k
views
When taking the fixed points commutes with taking the orbits
Let $G$ and $H$ be groups, both acting on a set $X$ on the left, in such a way that the two actions commute. (Equivalently, let $G \times H$ act on $X$.)
The set $\text{Fix}_H(X)$ of $H$-fixed ...
- 27.7k
5
votes
1
answer
889
views
A generalized Burnside's lemma
Let $G$ be a finite group acting on a set $X$, and let $S\subseteq G$ be a union of conjugacy classes. Then I believe I can prove:
$$ \sum_{[x]\in X/G} \frac{|G_x \cap S|}{|G_x|} = \sum_{g\in S} \...
- 66.8k
2
votes
1
answer
242
views
Products of groups with three generators
In this question it mentions how Jesse Douglas used a Zappa-Szep product to classify some finite groups with two generators, and others did the same thing with infinite cyclic groups. I've been ...
- 1,203
5
votes
2
answers
452
views
"geometric" description of the algebra of central functions on a Lie group
I am looking for a a description of the algebra of continuous central functions on a group, say a compact simple Lie group $G$, as the algebra of all continuous functions on a "nice" compact Hausdorff ...
- 2,201
9
votes
3
answers
675
views
Group extensions and actions on categories
Let G and H be two groups. There is a one-to-one correspondence between:
(i) an (isomorphism class of) extension of G by H, i.e. an exact sequence of group morphisms $1\to H\to E\to G\to 1$;
(ii) an ...
- 93
3
votes
1
answer
292
views
Special linear groups over function fields
Let $p$ be a prime number, and let $q$ be a finite power of $p$. Denote by $F_q$ the unique field with $q$ elements.
What is known about the structure and properties of $\mathrm{SL}_2(F_q[t])$ as ...
- 11.3k
4
votes
0
answers
187
views
Gaussian actions with no Bernoulli part
In an unrelated research project I came upon an example of a mixing unitary representation $\pi: \mathbb{F}_{\infty}\to B(\mathsf{H})$ of the free group on infinitely many generators, such that no ...
- 5,005
10
votes
4
answers
1k
views
Twist of a group Hopf-algebra
Let $G$ be a finite group with identity element $e$, and $C[G]$ the ring of complex-valued functions on $G$, with pointwise addition and multiplication. Then $C[G]$ is naturally a Hopf algebra, with ...
- 13.3k
27
votes
1
answer
2k
views
Strong group ring isomorphisms
Background/Motivation
Let $R$ be a commutative ring with unit. If $G$ is a finite (or in general, discrete) group, let $R[G]$ be the group $R$-algebra associated to $G$. The isomorphism problem for ...
- 23k
7
votes
1
answer
169
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 ...
- 11.3k
3
votes
2
answers
165
views
References about the matrix generators of the finite subgroups of the orthogonal group O(4)
"On Quaternions and Octonions" by Conway and Smith gives the classification of the finite subgroups of the orthogonal group O(4). I want to get the explicit matrix generators of the finite subgroups. ...
- 31
18
votes
0
answers
734
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 ...
- 11.3k
7
votes
1
answer
618
views
What version of the wreath product embedding theorem is actually stated in the famous paper of Kaloujnine and Krasner?
This question is inspired by Terry Tao's blog post and the comments there. I have always cited M. Krasner and L. Kaloujnine, "Produit complet des groupes de permutations et le problème d'extension de ...
- 38.6k
11
votes
1
answer
3k
views
Where can I easily look up / calculate (abelian) group cohomology?
For an example I'm trying to understand, I need to calculate some cohomology group of some $\mathbb Z$-module with coefficients in some other $\mathbb Z$-module (with no interesting actions). (In ...
- 54.6k
5
votes
2
answers
984
views
Automorphism Group of some Classical groups
Hi All,
I would like to know the Automorphism group of some simple classical groups, such as PSL(n,q) or some PSU or PSp groups. Could you please give me some recommended books or papers then? I ...
- 233
9
votes
1
answer
290
views
Calculations of nonabelian group cohomology of R^n
I am looking at $H^1(\mathbb{R}^n,G)$ where $G$ is a finite 2-group. I'm wondering if such things have been calculated. I'm afraid I can't say I know anything here, past the result that this ...
- 35.5k
1
vote
2
answers
557
views
Is there formula name and proof for this theorem ? [closed]
The formula answers: how many tuples $(\sigma_1,\sigma_2,...,\sigma_n)$ of elements of a given group G such that
(1) $\sigma_i\in C_i$ , where $C_i$ stands for conjugacy class.
(2) $\sigma_1\...
- 11
2
votes
2
answers
1k
views
Place stabilizers for the absolute Galois Group
Fix an algebraic closure, $\overline{\mathbb{Q}}$ for the rationals and consider the set, $B_p$, of all places of $\overline{\mathbb{Q}}$ over a fixed (possibly infinite) prime, $p$, of $\mathbb{Q}$. ...
- 1,049
3
votes
1
answer
708
views
vanishing higher cohomology group for property T group?
Given a countable discrete group $G$ with Kazhdan's property (T), consider $\mathbb{C}G$ or $l^2(G)$ as a left $G$-module, then we can consider the group cohomology,
Is it known that $H^n(G, l^2(G))=...
- 1,528
2
votes
1
answer
465
views
The first Betti number of a finite covering space of a closed 3-manifold
Let $\tilde{M}\to M$ be a finite covering of closed 3-manifolds.
Is it possible that $\beta_1(\tilde{M})-1> [\pi_1(M):\pi_1(\tilde{M})](\beta_1(M)-1)>0$?
Here, $\beta_1(X)=\dim_{\mathbb{Q}}H_1(...
- 541
8
votes
2
answers
840
views
Intersection of conjugates of subgroups in free groups
I am looking for a reference for the following
Fact 1: if $A$ and $B$ are finitely generated subgroups of infinite index in a finitely generated free group $F$ then there exists $f \in F$ such that $...
- 3,181
11
votes
1
answer
619
views
Analogues of the curve complex for Out(F)
Let $F$ be a finitely generated free group.
Question: Is there an authoritative survey of analogues of the curve complex for $\mathrm{Out}(F)$? If not, as seems likely, would a passing expert be ...
- 25k
7
votes
3
answers
577
views
Finitely presented groups which are not residually amenable
What are examples of finitely presented but not residually amenable groups?
Well, the examples that I want to have are simple f.p. groups as well as examples of non residually amenable groups arise ...
7
votes
1
answer
1k
views
Maximal compact subgroups of a semisimple Lie group are conjugate
I'm trying to go through the proof that all maximal compact subgroups of a semisimple Lie group $G$ are conjugate. I know that a possible proof follows the following steps:
Take one maximal compact ...
- 689
2
votes
1
answer
364
views
Totally aperiodic sequence
Let $A$ be a finite set. Let $A^k$ be the set of words in the alphabet $A$ of length $k$ and $A^*$ be the set of infinite words. I was looking for an element $a = \lbrace a_n \rbrace_{n \in \mathbb{N}}...
- 4,432
1
vote
2
answers
448
views
Lattices in general totally disconnected locally compact groups
Besides automorphism groups of trees and buildings, I was wondering if the lattices in general totally disconnected locally compact groups have been studied in the literature? I appreciate if you ...
user23860
6
votes
1
answer
435
views
Doubly primitive groups with simple socle
The classification of doubly transitive groups with simple socle is
known. A good account of such classification can be found for example
in this paper:
Cameron, Peter J. Finite permutation groups ...
- 3,078
6
votes
1
answer
900
views
Reconstruction Conjecture: Group theoretic formulation
As we read from wiki, informally, the reconstruction conjecture in graph theory says that graphs are determined uniquely by their subgraphs.
Is there a group-theoretic formulation of this conjecture?
...
- 2,855
1
vote
1
answer
362
views
Odd subgroup of $\mathrm{GL}(n,\mathbb{Z})$
The group $\mathrm{GL}(n,\mathbb{Z})$ acts on $(\mathbb{Z}/2\mathbb{Z})^n$ by right multiplication (the same kind of things can be done with left action). I denote by $H\subset \mathrm{GL}(n,\mathbb{Z}...
- 7,722
9
votes
2
answers
1k
views
Is it known if the absolute Galois group is "divisible"?
The definitions of a divisible group that I have seen all seem to assume abelian is an a priori property of the group. My question is as to whether or not it is known that--given a non-torsion element ...
- 1,049
2
votes
0
answers
96
views
A kind of cancellation ; exchange problem for groups
For which $(m,n,k,l) \in (\mathbb N\cup \{0\})^4$ , with $m\le n ; k\le l$ , does there exist a group $G$ with a finite subnormal series with torsion-free Abelian quotients such that $G \times \mathbb ...
user111524
2
votes
0
answers
130
views
Existence of a transfinite sequence of abelian groups having a strange property
I am studying a paper which uses the following lemma. The context is irrelevant, as the lemma is only used as a technical trick and has no pointer to a reference or hint in the proof but its link to ...
- 9,111
12
votes
3
answers
552
views
Estimate on currents in Cayley graphs
Take a Cayley graph $\Gamma$ (thought of as an electrical network with all edges having equal resistance) and break one edge $e$ and put a battery there. (Assume the graph has only one end* so that ...
- 4,432
11
votes
2
answers
4k
views
Orders of automorphism groups of p-groups
There is a theorem that says that if $p$ is a prime and $G$ is a $p$-group with $|G| = p^{n}$, $|Aut(G)|$ divides $\Pi_{k=0}^{n-1} (p^{n}-p^{k})$.
This theorem is sharp, since $\Pi_{k=0}^{n-1} (p^{n}-...
- 3,645
11
votes
1
answer
1k
views
Strong Atiyah conjecture
Who introduced the Strong Atiyah Conjecture?
Recall that the conjecture says the following. Let $G$ be a group, $A$ a $n\times n$-matrix over ${\mathbb Z}G$. We view $A$ as a bounded operator $l^2(...
user6976
11
votes
1
answer
1k
views
When is an HNN-extension finitely presented?
Let $G=\langle H, t; K^t=K^{\prime}\rangle$ be an HNN-extension of $H$, with $t$ inducing the isomorphism $\phi: K\rightarrow K^{\prime}$. I was wondering if the following question can be answered, ...
- 2,821
2
votes
0
answers
414
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 ...
- 4,432
3
votes
0
answers
62
views
Torus in the small Ree group ${}^2G_2$ over an infinite field
In “Simple group of Lie type” by R. W. Carter there is a remark (after Theorem 13.7.4):
It is not known whether $H^1$ coincides with the set of $\sigma$-invariant elements of $H$ if $\mathfrak{L}$ ...
- 3,186
3
votes
1
answer
309
views
Intersection of maximal subgroups of PSL(2,q)
Let $G := PSL(2,2^r)$, and let $M$ be a maximal subgroup of $G$ isomorphic to $PSL(2,2^s)$. I need to compute $H := M \cap M^g$ for $g \in G-M$. It seems to me that $|H|$ must be $2^r, 2^r\pm 1$ or $...
- 307
5
votes
3
answers
645
views
Reference request for the number of Sylow p-subgroups
Let $G$ be simple group of Lie type or Alternating group. I need reference for find the number of Sylow $p$-subgroup $G$ for every $p$. Thanks a lot.
- 221
3
votes
0
answers
493
views
Short exact sequences for amalgamated free products and HNN Extensions
I asked this question on math stackexchange (see here) but didn't get any answer so I thought I would post it here too:
If $A$ and $B$ are groups we have the following short exact sequence:
$$ 0 \to [...
- 721