Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
1,094 questions
6
votes
0
answers
255
views
Homotopy quotient of groups
Suppose $0\to A \stackrel{\iota}{\to} B \stackrel{\pi}{\to} C \to 0$ is a short exact sequence of groups.
We have an induced map $k[\iota] : k[A] \to k[B]$ of group algebras over a field $k$.
What ...
- 1,382
6
votes
0
answers
453
views
Does the Approximation Property (AP) pass to quotients by amenable subgroups?
Given a countable group $G$ with the AP, and an normal, amenable subgroup $N$ of $G$, does $G/N$ have the AP?
In particular, does there exist a group $G$ with the AP and a surjective group ...
- 3,497
6
votes
0
answers
536
views
Groups whose finite index subgroups are isomorphic
I am looking for examples of the following situation:
$G$ is an infinite group. Every two finite index proper subgroups of $G$ are isomorphic.
The only examples that I have now are (1) $\mathbb{Z}^...
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
338
views
Semisimple compact Lie group topologically generated by two finite order elements
Edit: I'm specializing this question to the compact case I'll ask about the noncompact case as a new question.
Let $ G $ be a compact connected semisimple Lie group.
Do there always exist two finite ...
- 4,081
6
votes
1
answer
213
views
Is there a subgroup of a non-abelian $p$-group $G$ with a large nilpotency class?
Let $G$ be a non-abelian $p$-group ($p\ne2$). Does there exist a group $H\subset G$ such that both 1, 2 are satisfied?
$|H| = |G|/p$.
$c(H)\geq c(G) - 1$.
- 291
6
votes
2
answers
693
views
Groups whose centralisers are finite
Let $G$ be an infinite group such that the centraliser of any non central element is finite (and bounded).
Is there any structure theorem known about $G$ ?
Such a group seems to be at the other ...
- 1,555
6
votes
0
answers
430
views
More on Moonshine for the Thompson group and weakly holomorphic weight one half modular forms
This question is a follow-up to
Monstrous Moonshine for Thompson group $Th$?
and is based on various comments to that question, in particular S. Carnahan's mention of the
connection to known ...
- 5,546
6
votes
1
answer
444
views
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| \...
- 156k
6
votes
2
answers
353
views
The Tits classes of simply connected simple real groups
Let $G$ be a simply connected semisimple group over a perfect field $k$ (at the moment I am interested in the case $k=\mathbb R$).
Then $G$ is an inner form of a quasi-split $k$-group $G_{\rm qs}$:
...
- 14.2k
6
votes
2
answers
2k
views
If $G \times G \cong H \times H$, then is $G \cong H$? [duplicate]
Let $G,H$ groups. Suppose that $G \times G \cong H \times H$. Then is necessarily $G \cong H$?
I know that if $G,H$ are finite groups then it is true (you show that the map $FinGrp \rightarrow \...
- 69
6
votes
1
answer
435
views
Dynkin diagram of the centralizer of a semisimple element in a Levi subgroup
Let $G$ be a connected reductive group over an algebraically closed field and consider a semisimple element $s \in G$ and let $L$ be a Levi subgroup containing $s$.
My question is about the two ways ...
- 309
6
votes
1
answer
1k
views
Discrete central subgroup of a connected Lie group is finitely generated
Every discrete central subgroup of a connected Lie group is finitely generated.
This result was alluded to without comment in a book I was reading (Lie Group Actions in Complex Analysis by D. ...
- 675
6
votes
1
answer
564
views
Split powers of the multiplicative group of a field
Let $K$ be a field, $K^\times$ its multiplicative group and $I$ an infinite set. Is then $(K^\times)^{(I)} \subseteq (K^\times)^I$ a direct summand? If not, is it possible to characterize the fields ...
- 63.1k
6
votes
0
answers
618
views
Duality between conjugacy classes and irreducible characters for finite monoids?
Qiaochu's answer to this question suggests that the proper way to view the bijection between conjugacy classes and irreducible complex representations of a finite group is via a duality. My question ...
- 38.6k
6
votes
1
answer
355
views
Do doubly-transitive actions give rise to indecomposable representations for infinite groups?
This is a follow-up to this question.
Let $G$ be a group acting doubly-transitively on a set $X$. Then the vector space $V_X$ of functions $f\colon X\to\mathbb C$ with finite support such that $\sum_{...
- 3,054
6
votes
0
answers
190
views
The highest degree of a polynomial on a finite group
This question is motivated by the comments and the answer to this MO-question.
First let us recall some definitions.
A function $f:X\to X$ on a group $X$ is called a polynomial if there exists $n\in\...
- 41.9k
5
votes
0
answers
298
views
What are the matrix coefficients associated with the irreducible representations of compact real linear algebraic groups?
What are the matrix coefficients associated with the irreducible representations of a compact real linear algebraic group $G$?
Peter-Weyl tells us that $L^2(G)$ is the (closure of) $\bigoplus_\pi A_{\...
- 2,071
5
votes
2
answers
375
views
a balanced presentation of a cyclic-by-cyclic group?
Let $p>2$ be a prime, $C_p$ be the additive group of integers mod $p$. Then the multiplicative group $\{1,...,p-1\}$ of units in the field $Z/pZ$ is cyclic of order $p-1$, it acts on $C_p$ by left ...
user6976
5
votes
0
answers
351
views
Adjoint identity on finite nilpotent groups
Let $G$ be a finite nilpotent group. Consider the following (adjoint) identity from [BuPa, Theorem 9.6]:
$$\left(\prod_{\chi \in \mathrm{Irr}(G)} \frac{\chi}{\chi(1)} \right)^2 =\frac{|Z(G)|}{|G|}\...
5
votes
1
answer
617
views
Fixed points on boundary of hyperbolic group
Let G be a word-hyperbolic group with torsion and let ∂G be its boundary. Do there exist criteria that imply that all non-trivial finite order elements of G act fixed-point freely on ∂G?
- 245
5
votes
1
answer
232
views
Converse of Schreier theorem
I know that every subgroup of a free group is free (Schreier theorem).
I'm wondering that a (non-trivial) converse is true, that is, if every proper subgroup of an infinite group $G$ is free, then $G$ ...
- 125
5
votes
3
answers
359
views
Modular reductions of simple characters
Given a (splitting) $p$-modular system $(K, \mathcal{O}, k)$ for a finite group $G$, any given simple character $\chi$ is afforded by some $KG$ module $V_\chi$, and there in general many non-...
- 298
5
votes
2
answers
524
views
How large can abelian subgroups of class 2 nilpotent groups or simple groups be?
If $G$ is a finite simple group then is it true that an abelian subgroup $H$ of $G$ of maximal order has order $|H| < |G|^{\frac{1}{3}}$? If so, could you please point me to a reference for this, ...
- 181
5
votes
1
answer
308
views
Quotient groups obtained by quotienting $G^n$ by $G^{n-1}$
Notation: For a group $G$, we write $G^n$ to denote the $n$-fold direct product of $G$ with itself.
Problem set up:
Consider, for a finite group $G$, and $n > 1$, the set $Q(G)_n$ of all ...
- 6,223
5
votes
3
answers
1k
views
Action of a profinite group
Let $G$ be a finitely generated profinite group, $p$ a prime number. Put $$ V = \prod_{i \in I} \mathbb{Z}_p$$ a (profinite) group equipped with the product topology (for convenience, $I$ may be ...
- 11.3k
5
votes
2
answers
535
views
Non-trivial extension of representations have same central character
Let $\pi_1, \pi_2$ be two irreducible complex representations of $G=\mathrm{GL}_2(\mathbb{Q}_p)$ and assume that there exists a non-split extension $0\to\pi_1\to \pi\to\pi_2\to0$ of representations ...
user515481
5
votes
1
answer
1k
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 ...
- 591
5
votes
0
answers
171
views
Spectral sequence construction of Euler class of group extension
Let $A$ be an abelian group equipped with an action of a group $G$ and let
$$1 \longrightarrow A \longrightarrow \Gamma \longrightarrow G \longrightarrow 1$$
be an extension of group inducing the ...
- 51
5
votes
0
answers
169
views
In the literature on infinite graphs, are there results on "periodizable" graphs?
Let $G=(V,E)$ be a connected countably infinite $k$-regular simple graph (no loops or multiple edges). For $A$ a finite subset of $V$, let me denote by $G_A=(A,E_A)$ the induced subgraph with vertex ...
5
votes
1
answer
729
views
Finitely generated solvable groups all of whose abelian normal subgroups are finite
Is there a classification for infinite finitely generated solvable groups all of whose abelian normal subgroups are finite?
I mean by classification something like presentation.
Edited: Is there an ...
- 3,738
5
votes
1
answer
384
views
Which groups have undetectable third U(1)-cohomology?
Let $G$ be a finite group. A categorical Schur detector for $G$ is a set $\mathcal{S}$ of proper subgroups $S \subsetneq G$ such that the total restriction map
$$ \mathrm{rest}_{\mathcal{S}} : \mathrm{...
- 54.6k
5
votes
1
answer
311
views
Projective representations of a finite abelian group
Projective representations of a group $G$ are classified by the second group cohomology $H^2(G,U(1))$. If $G$ is finite and abelian, it is isomorphic to the direct product of cyclic groups
$$
G\cong ...
- 813
5
votes
1
answer
294
views
Words which are not inverted by any endomorphism
Let $w$ be a word in a free group $F_2$ of two generators $x_1, x_2$ such that there does not exist any endomorphism of free group which takes $w$ to $w^{-1}$. Let $w_1, w_2$ be two words in the same ...
- 355
5
votes
3
answers
448
views
Variation of centraliser in $\operatorname{GL}(n,\mathbb{Z})$
$\DeclareMathOperator\GL{GL}$Let $n$ be a positive integer $\geq 2$. The setting is that $K \in \GL(n,\mathbb{Z})$, and people are interested in understanding the centralizer:
$$
C(K)=\{ B \in \GL(n,\...
- 145
5
votes
1
answer
909
views
Finding a basis for the (linear combinations) span of a matrix group, efficiently?
I have an algorithm whose bottleneck is the following task:
Let $\mathbb{F}$ be a finite field.
Given a set of $k$ invertible matrices $g_1,\dots,g_k\in GL_n(\mathbb{F})$, let
$G=\langle g_1,\dots,...
- 3,104
5
votes
2
answers
1k
views
Special automorphisms of extraspecial groups
Let $G$ be an extraspecial group of order $p^{2r+1}$ (where $p$ is an odd prime), and let $V$ be a faithful representation of $G$. Consider the normal subgroup $H$ of $Aut(G)$ consisting of all ...
- 51
5
votes
1
answer
138
views
Groups (not) quasi-retracting onto $\mathbb{Z}$
Let $X$ and $Y$ be metric spaces, with metrics $d_X$ and $d_Y$ respectively. A function $f\colon X\to Y$ is coarse Lipschitz if there exist constants $C,D>0$ such that $d_Y(f(x),f(x'))\le C d_X(x,x'...
- 3,740
5
votes
0
answers
299
views
A class 3 group of order 243
Let G be a group of order $243=3^5$. We denote by $(G_i)$ its lower central series and assume that $G$ has class $3$ and that $|G:G_2|=|G_3|=9$. We assume moreover that the cubing map factors as a (...
user76083
5
votes
1
answer
342
views
Relations between boundaries of groups acting on hyperbolic spaces with WPD elements
Let $(X,d)$ be Gromov-hyperbolic space and let $\Gamma$ be a finitely generated group acting on $\Gamma$ by isometries. Recall the following two definitions.
Say that the action is acylindrical if ...
- 2,090
5
votes
2
answers
441
views
Reference Request: Derived group of $\mathscr R_u(B)$
Let $G$ be a connected, reductive group over an algebraically closed field $k$. Let $B$ be a Borel subgroup with maximal torus $T$ and unipotent radical $U$. Let $\Phi^+ = \Phi(B,T)$ and $\Delta$ ...
- 6,180
5
votes
0
answers
439
views
Generalized pure braid groups
Generalized braid groups are by definition obtained from finite Coxeter groups after removing some of its relations. For example the Coxeter group $H_3=< a_1, a_2, a_3 | a_i^2=1, a_1a_3=a_3a_1, ...
- 59
5
votes
3
answers
697
views
Is it known whether there is a non-coherent group with a two- or three-relator presentation?
To the best of my knowledge, whether 1-relator groups are coherent is still an open question. The group $F_2 \times F_2$ (where $F_2$ is the free group of rank $2$) is well-known to be non-coherent ...
5
votes
2
answers
489
views
The generalized word problem on groups
Given some group $G$ that is generated by $a$ and $b$, each of which has infinite order, and some free subgroup $N$ generated by $a^k$ and $b^k$, is there any algorithm that tells me if some $x \in G$ ...
- 469
5
votes
2
answers
292
views
Field with one element look at counting index-$n$ subgroups in terms of Homs to $S_n$, generalization to $F_{1^k}$?
Main idea shortly: As we discussed recently MO272045, there is beautiful fomula which
counts index-n subgroups in terms of homomorphisms to $S_n$.
Let me give "field with one element" interpretation ...
- 24.9k
5
votes
1
answer
227
views
Generalized identities of (soluble) groups
Let $G$ be a group. Let us say that $G$ satisfies a generalized identity of degree $n$ if there exist $a_1,a_2,\dots a_n \in G$ such that
$$x^{a_1}x^{a_2}\dots x^{a_n}=1,$$
for all $x\in G$.
Assume ...
5
votes
3
answers
450
views
Restriction map between spaces of automorphic forms
Hello,
Let $H \subset G$ be reductive groups defined over $\mathbb{Q}$. I consider the spaces of automorphic forms of $G$ and $H$. One has a restriction map from the space of automorphic forms of $G$ ...
- 653
5
votes
1
answer
820
views
Maximal subgroups of semisimple Lie groups
The problem of finding and classifying the maximal subgroups of simple Lie groups like $SU(3)$ is well known and solved in the literature. What about maximal subgroups of semisimple groups like $SU(3) ...
user22139
5
votes
2
answers
976
views
Root system automorphisms as inner automorphisms of extended Chevalley group
For each automorphism $\sigma$ of a root system $\Phi$ there is a unique automorphism of the Chevalley group $G(\Phi,R)$ such that $\sigma(x_\alpha(t))=x_{\sigma\alpha}(t')$. While conjugating by ...
- 3,186
5
votes
3
answers
501
views
Can we classify all finite 2-generated groups $G$ such that if $x,y$ generate $G$, then so does $x,yxy^{-1}$?
Can we classify finite 2-generated groups $G$ satisfying the following property:
For any pair $x,y$ which generate $G$, the pair $x,yxy^{-1}$ also generates $G$.
By the comments, no nontrivial ...
- 10.7k