All Questions
Tagged with reference-request gr.group-theory
700 questions
6
votes
1
answer
464
views
Adjoint orbits of a finite group of type $G_2$
Let $q=p^\alpha$ be a prime power and $k=\mathbb{F}_q$. Let $G\subseteq \mathrm{GL}_N(k)$ be a simple finite group of Lie type, with root system of type $G_2$, and let $\mathfrak{g}\subseteq \mathfrak{...
- 993
7
votes
2
answers
669
views
Élie Cartan's paper "Les groupes réels simples, finis et continus" of 1914
Question 1.
Does Élie Cartan's paper
Les groupes réels simples, finis et continus,
Ann. Sci. École Norm. Sup. (3) 31 (1914), 263–355
contain a classification of $\Bbb C$-linear involutions of simple ...
- 14.2k
8
votes
0
answers
137
views
Group presentations where discarding generators always yields a subgroup
Consider a group presentation $ \left< G= \left\lbrace \text{generators}\right\rbrace \, \middle|\, R = \left\lbrace \text{relators}\right\rbrace \right>$ (no finiteness assumptions). Given $S\...
- 1,005
5
votes
3
answers
851
views
What are some examples of non-commutative $\mathbb{Q}$-monoids and/or $\mathbb{R}$-monoids?
Definition 0. Let $R$ denote a commutative semiring with $0$ and $1$. By an $R$-monoid, I mean a monoid $M$ equipped with an action $R \times M \rightarrow M$ denoted $r,m \mapsto m^r$, satisfying the ...
- 3,793
4
votes
0
answers
320
views
Finite subgroup of $\mathrm{SO}(4)$ which acts freely on $\mathbb{S}^3$
Let $\Gamma$ be a finite subgroup of $\mathrm{SO}(4)$ acting freely on $\mathbb{S}^3$. It is known that all such $\Gamma$ can be classified.
Is there any characterization of $\Gamma$ such that $\Gamma$...
- 1,441
1
vote
0
answers
287
views
Characters of upper triangular matrices over finite field - reference request
Let $B_n$ be the group of upper matrices and $U_n$ the subgroup of unipotent upper triangular matrices. I would like some references which discusses complex character theory of $B_n(\mathbb{F}_q)$ for ...
- 2,751
3
votes
1
answer
137
views
Pairs of elements in $F_n$ with distinct translation lengths
Let $F_n$ be a free group of rank n and consider all possible non-degenerate length functions on $F_n$.
Could I be directed to a reference that give two non-trivial non-power-conjugate elements $g,h \...
- 1,033
7
votes
0
answers
405
views
How can I get my hands on McKay's "Finite p-groups" lecture notes?
How can we find Susan McKay's "Finite $p$-groups" lecture notes?
The notes I'm talking about are these.
I emailed Peter Cameron, but he has since moved to a different university, and has no ...
- 4,425
9
votes
1
answer
458
views
Fuchsian groups and Eichler's result
Let $G$ be a Fuchsian group of first kind contained in $\text{PSL}_2(\mathbb{R})$. A result of Eichler says, there exists a finite set $S\subset G$ such that any $\gamma$ in $G$ can be written as a ...
- 521
5
votes
1
answer
597
views
Can every cancellative invertible-free monoid be embedded in a group?
A monoid is invertible-free if $xy=1$ implies $x=y=1$ for all $x,y$.
Question: Can every cancellative invertible-free monoid be embedded in a group?
I'm fairly sure that a quotient of the free product ...
3
votes
0
answers
296
views
Amenability, growth and asymptotic dimension
I recently found this question on MSE, relating growth of groups with whether they are amenable, elementary amenable or not. I would like to know if there is an extra relation to finite or infinite ...
- 267
5
votes
3
answers
230
views
Maximal subgroups of odd index in $\mathrm{PSL}(3,q)$
Let $G = \mathrm{PSL}(3,q)$ for $q$ odd. I am trying to understand a question that involves understanding the subgroups that contain a Sylow $2$-subgroup, and in particular, are subgroups of odd index ...
- 366
2
votes
0
answers
124
views
Heap torsors and dual objects
Suppose that $G$ is a group and $P:G\rightarrow G$ is a permutation of $G$. We would now like to apply a forgetful functor to the algebra $(G,\cdot,P,^{-1},e)$. Whenever $g\in G$, let $gP$ be the ...
3
votes
0
answers
205
views
Status of RFD groups and $C^*$-algebras
Motivated by this question and its great answers, I become very curious to know what do we know about RFD (residually finite dimensional) groups and $C^*$-algebras, e.g. do we know how these ...
- 1,586
10
votes
1
answer
602
views
hyperbolic quotient of hyperbolic group
I have a memory of hearing about a result (or perhaps a conjecture), possibly due to Gromov, that, if $G$ is a hyperbolic group and $g \in G$ has infinite order, then the quotient group $G/\langle (g^...
- 37.4k
13
votes
3
answers
2k
views
Which groups are LERF?
A finitely generated group $G$ is called LERF if every finitely generated $H \leq G$ is closed in the profinite topology on $G$ (equivalently, there is a family of finite index subgroups of $G$ ...
- 11.3k
8
votes
1
answer
396
views
Which group do two generic $2\times 2$ triangular matrices generate?
Let $A,B$ be two generic (in particular invertible) $2\times 2$ upper-triangular complex matrices. They generate a countable group $G$, the commutator subgroup of $G$ is abelian. Are there other ...
- 109k
2
votes
0
answers
161
views
Monotonicity of the cycle index polynomial under restriction
The cycle index (polynomial) of the symmetric group $\mathfrak{S}_n$ is given by the formula:
$$Z(\mathfrak{S}_n)(x_1,\dots,x_n)=\sum_{1j_1+2j_2+\cdots+nj_n=n}\prod_{k=1}^n\frac{x_k^{j_k}}{k^{j_k}j_k!}...
- 43.2k
2
votes
1
answer
217
views
A variation of closed-subgroup theorem
$\DeclareMathOperator\SO{SO}$Recall that the closed-subgroup theorem (Wikipedia link) says that a closed subgroup of a Lie group is a Lie group.
I am pretty sure that this theorem should have a "...
- 14.3k
4
votes
2
answers
412
views
Generators of the orthogonal group of a quadratic form in odd dimension in characteristic 2
In characteristic not $2$, the Theorem of Cartan-Dieudonné states:
[Grove, Theorem 6.6]: Let $q$ be a nondegenerate symmetric quadratic form of dimension $n$ in characteristic not $2$. Then every ...
- 168
13
votes
2
answers
414
views
Is every finite-order unimodular matrix conjugate to a $0,1,-1$ matrix?
Problem. Given a matrix $A\in\mathrm{GL}(n,\mathbb{Z})$ such that $A^k=1$ for some $k\geq 1$, is there a matrix $g\in\mathrm{GL}(n,\mathbb{Z})$ such that $gAg^{-1}$ has only $0$, $1$, and $-1$ as ...
- 23.3k
1
vote
1
answer
435
views
Direct proof that free groups are sofic
I am looking for a reference (or a simple proof) of the fact that a free group is sofic. The preferred dynamical definition of a sofic group seems to be that
there is a sequence of finite sets $V_n$ ...
- 23.2k
3
votes
3
answers
714
views
Cohomology of elementary abelian $p$-groups, i.e. $H(G,{\mathbb F}_p)$ with $G\cong{\mathbb F}_p^r$
I have two questions.
$\bf 1.$ First, a reference request. Let $G\cong{\mathbb F}_p^r$ for some integer $r\geq 0$ and let $V=G^*={\rm Hom}(G,{\mathbb F}_p)$. Then $(H(G,{\mathbb F}_p),+,\cup )$ is a ...
10
votes
3
answers
431
views
A malnormal embedding theorem?
Let $Q$ be a recursively presented group. Is it possible to embed $Q$ into a finitely presented group $G$ such that the image of $Q$ is malnormal in $G$?
Note that a subgroup $H$ of $G$ is malnormal ...
- 2,821
9
votes
1
answer
738
views
Gromov hyperbolic groups which are solvable are elementary
I have read on wikipedia that a Gromov hyperbolic group which is solvable is elementary (i.e. virtually cyclic). Where can I find a proof of this fact?
There is a proof of a similar fact in Bridson-...
- 291
6
votes
1
answer
186
views
Endomorphism ring of trivial source modules for abelian p-groups
Bernhard Böhmler (who is also on MO) and myself had the following idea:
Let $G$ be a finite group and $k$ a field of characteristic $p$ (algebraically closed when it is needed) such that $p$ divides ...
- 26.5k
3
votes
1
answer
244
views
Finitely-generated conjugation action on a subgroup that is not normal... what is that?
If $H \lhd G$, then $G$ acts on $H$ by conjugation. I need to talk about this action but in a situation where $H$ is not (necessarily) normal. When $H \leq G$, there is a "partial action" of ...
- 6,652
29
votes
2
answers
1k
views
Quillen + construction for finite groups
Is there an example of two non isomorphic finite groups $G$ and $H$ such that $BG^{+}$ is homotopy equivalent to $BH^{+}$ ?
- 1,629
4
votes
0
answers
552
views
Lattices of $\mathbb{R}^s \ltimes_\varphi \mathbb{R}^k$
Edit: Thoughts updated (22/3/2021).
I've come across with the following problem.
Let $G=\mathbb{R}^s \ltimes_\varphi \mathbb{R}^k$ where $\varphi:\mathbb{R}^s\to \mathrm{Aut}(\mathbb{R}^k)=\mathsf{GL}(...
5
votes
1
answer
175
views
Looking for a BAMS text about the group with commutation relations defined using meaningful words
What I definitely remember is that I saw a description of the following in the Bulletin of the American Mathematical Society, sometime in eighties (or maybe nineties?)
One considers the group ...
- 18.4k
10
votes
3
answers
725
views
Reduction mod $n$ of symplectic group
Let $g,n$ be positive integers, is there a reference that $\mathrm{Sp}(2g,\mathbb{Z})\to\mathrm{Sp}(2g,\mathbb{Z}/n\mathbb{Z})$ is surjection?
The only reference I could find is lemma 5.16 in Deligne–...
user39380
3
votes
2
answers
211
views
Classification of finite simple groups with abelian Sylow 2-subgroups
In this MathSE question,
classification of finite simple groups with Abelian Sylow 2-subgroups,
credit is rightly given to John Walter. But in the introduction to his paper, Walter explicitly states ...
- 35
2
votes
0
answers
110
views
Moment of the hitting measure of a subgroup
Given a [finitely generated] group $G$ and a finite generating set $S$, a measure $\mu$ will have finite $\alpha$-moment if $\sum_{g \in G} \mu(g) |g|_S^\alpha$ is finite (where $|g|_S$ is the word ...
- 4,432
7
votes
1
answer
143
views
Is there a pseudofinite group with a quantifier-free instance of the order property?
Recall that a group $G$ is pseudofinite if every first-order sentence $\varphi$ (in the language of groups) satisfied in $G$ is also satisfied in some finite group. Also recall that an instance of the ...
- 12.5k
8
votes
2
answers
617
views
Relative/acylindrical hyperbolicity of free-by-cyclic groups
Is this statement true?
Let $\mathbb{F}$ denotes a finitely generated free group, $\Phi$ an automorphism of $\mathbb{F}$ and $\varphi$ its image in $\mathrm{Out}(\mathbb{F})$.
If $\varphi$ is ...
- 1,755
9
votes
2
answers
762
views
Solutions of $x^d=1$ in the symmetric group
L Moser and M Wyman, On solutions of $x^d = 1$ in symmetric groups, Canad. J. Math., 7 (1955), pages 159-168, explored asymptotic behavior of the cardinality of such permutations:
$$f_d(n):=\#\{\pi\in\...
- 43.2k
10
votes
3
answers
913
views
Explicit free subgroup in Thompson's group $V$
R. Thompson introduced three groups $F\subset T\subset V$. The question concerning amenability of $F$ is still unanswered and has attracted much attention. I have read that Thompson group $V$ contains ...
- 743
2
votes
0
answers
114
views
understanding the definition of subgroup of the Grothendieck-Teichmuller group
Definition. Let $\widehat{G T}^{1}$ be the set of elements $f$ in the derived subgroup $\hat{F}_{2}^{\prime}$ of $\hat{F}_{2}$ such that $x \mapsto x$ and $y \mapsto f^{-1} y f$ extends to an ...
- 204
1
vote
0
answers
175
views
Cochains with multilinear differentials
Let $G$ be a group and let $M$ be a $G$-module. We denote by $(C^*(M,G),d)$ the complex of inhomogeneous cochains, i.e. $C^n(G,M)=M^{G^n}$.
We say that a cochain $a\in C^n(G,M)$ is multilinear if it ...
8
votes
3
answers
559
views
Reference for tetrahedral Coxeter group
Let $G$ be the group with 4 generators, each of order 2, such that the product of any 2, say $ab$, has order 3 (i.e., $ababab=e$).
That is, this is an infinite reflection group with Coxeter diagram a ...
- 83
16
votes
4
answers
1k
views
Origin of group theory problem (bound on number of Sylow subgroups)
This problem (prove that the number of Sylow subgroups of a finite group $G$ is bounded by $\frac{2}{3}|G|$) posted on MSE proved rather difficult to solve. The OP has been silent about where the ...
- 2,168
6
votes
0
answers
234
views
Nascent formal group law
$\DeclareMathOperator\FGL{FGL}$The formal group law (cf. Wikipedia, Ex. 1.6 of nLab, Hazewinkel) derived from an analytic function or formal series $f(x) = x + a_2 x^2 + a_3 x^3 + ...$ and its formal ...
- 10.5k
12
votes
4
answers
2k
views
Efficient presentations for finite groups
A finitely presented group which has more generators than relations has an infinite abelianization and so is an infinite group. Therefore, for a finite group, all presentations must have at least as ...
- 3,645
9
votes
4
answers
2k
views
Commuting matrices in GL(n,Z)
Suppose $M$ is a "hyperbolic" matrix in $GL(n,\mathbb Z)$, i.e., that its characteristic polynomial $p$ is irreducible over $\mathbb Z$ and has no roots of modulus 1.
Is there a closed description ...
- 2,135
5
votes
2
answers
1k
views
Malcev's paper "On a class of homogeneous spaces" in English
I am struggling to find the English translation of Malcev's paper "On a class of homogenous spaces" providing foundational material for nil-manifolds. To be precise this paper: Malcev, A. I. On a ...
- 51
5
votes
1
answer
201
views
An Indepth Look at Isoperimetry in the Cayley Graph Generated by All Transpositions
Let $\Omega_n$ denote the symmetric/permutation group on $n$ objects.
Let $T_n \subseteq \Omega_n$ denote the set of transpositions.
Drop the $n$-subscripts.
Define the Cayley graph $G = (\Omega, E)$ ...
- 560
4
votes
1
answer
446
views
What is a "cusp" ("кусок") in relation to Guba's embedding theorem?
I'm confused by the definition of a "cusp" as found in
V.S. Guba, Conditions for the embeddability of semigroups into groups, Math. Notes 56 (1994), Nos. 1-2, 763-769 (link).
In the words of Mark ...
- 10.5k
3
votes
1
answer
141
views
The stabilizer of a pair of points in the acylindrically hyperbolic group is either finite or virtually cyclic
Given a group $G$, suppose $G$ admits a non-elementary acylindrical action
on a Gromov hyperbolic space $S$.
I heard that stabilizer of a pair of points on $\partial S$ in the acylindrically ...
- 199
9
votes
1
answer
1k
views
What is the outer automorphism group of $\operatorname{SL}(2,\mathbb{F}_q)$?
I'm looking for a reference for a description of the outer automorphism groups of $\operatorname{SL}(2,\mathbb{F}_q)$ for $q = p^n$.
I'm sure such a thing must exist somewhere, but I'm having trouble ...
- 7,527
8
votes
1
answer
513
views
Equivariant (co)homology of flag manifolds, convolution algebra and nil hecke algebra?
For a complex reductive group $G$ and its Borel subgroup $B$, it seems to be well-known that the equivariant homology group $H^G_*(G/B\times G/B)$ forms a nil-Heck algebra
$$NH=\Bbbk[y_i,\partial_{j}]...
- 719