All Questions
Tagged with reference-request gr.group-theory
700 questions
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 ...
5
votes
0
answers
266
views
Character tables of finite groups and isomorphism
I'd like to ask the following question:
Let $G$ and $H$ be finite groups.
Is there a useful criterion involving the ordinary character table which assures that $G$ and $H$ are isomorphic as groups?...
- 1,755
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
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
446
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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
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
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
2
votes
2
answers
419
views
Where can I find a table of the exponents of the sporadic groups?
Is there a table showing Sporadic Groups and their exponents, and, perhaps, other basic properties.
In particular, I'm interested in what the exponent of the Monster Group is. (Obviously the order is ...
- 373
11
votes
0
answers
382
views
Ascending chain condition for 1-element normal closures in a free group
Let $F$ be a free group of finite rank. Does $F$ satisfy the ascending chain condition on normal subgroups each of which is a normal closure of one element?
In other words, can there exist elements $...
- 3,181
1
vote
0
answers
126
views
Algebraic structures on graphs
There are many algebraic structures linked to graphs.
For example one can find zero divisor graphs $[1]$, $[2]$ and many other graphs.
Does there exist any survey paper which characterizes all the ...
- 444
11
votes
0
answers
345
views
Status of questions in "Group Actions on $\mathbb{R}$-trees"?
Culler and Morgan's "Group Actions on $\mathbb{R}$-trees" lists four questions at the end of the introduction. A few have been famously resolved by work of Rips, Bestvina–Feighn and others.
I'm ...
- 1,996
2
votes
0
answers
133
views
Looking for the multiplicity-free paper by N.Inglis
I'm looking for the paper "Multiplicity-free permutation characters, distance-transitive graphs and classical groups, PhD Thesis, University of Cambridge, 1986" by Nicholas Francis John Inglis. The ...
- 9,501
0
votes
0
answers
46
views
Generalizing CIT-groups to odd case
A CIT-group is a group such that the centralizer of any involution is a 2-subgroup. The structure of these groups is known from the works of Suzuki and others.
Here is my question: has the odd case ...
- 307
2
votes
0
answers
89
views
Name for a probability density ''symmetrized'' by a permutation group?
Let $p$ be a probability density function over random variable $X$, and $G$ a compact permutation group over the outcomes of $X$. For each $g\in G$, let $p_g$ indicate the probability density ...
- 695
1
vote
0
answers
76
views
Nomenclature: does this coset space have a name?
in my work I tripped on a specific coset space and before starting thinking about it by myself, I wanted to check the literature. However, I do not know if the object has a name (which makes ...
- 111
2
votes
0
answers
135
views
English translation of Fouxe-Rabinovitch paper
Is there somewhere an english translation of Fouxe-Rabinovitch's papers
"D. I. Fouxe-Rabinovitch, Uber die Automorphismengruppen ¨
der freien Produkte. II, Rec. Math. [Mat. Sbornik] N.S., 1941,
...
- 257
2
votes
0
answers
137
views
Time complexity of randomized algorithm: right-multiplying by random elements $z_i$ from a group $H$ to achieve $H$-invariance
Note: This question was inspired by a related question about the Quantum Merlin Arthur (QMA) complexity class on Quantum Computing Stack Exchange. I was deliberating whether to ask this on CS Theory ...
- 269
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
2
votes
1
answer
151
views
Topological analogue of an FC group?
By definition, a group is FC if all its conjugacy classes are finite.
Has anything been published about a generalization of the FC property for topological groups?
- 51
3
votes
0
answers
94
views
Clifford correspondence(s) from Fong-Reynolds theorem
The Fong-Reynolds theorem states a certain relationship between blocks of a normal subgroup $N\unlhd G$ and blocks of $G$, sometimes called the "Clifford correspondence for blocks". If one phrases ...
- 9,650
2
votes
0
answers
115
views
The structure of $PSL_2$ over the p-adic integers
As is well known, the group $\mathrm{PSL}(2,\mathbf{Z})$ is isomorphic to the free product of two cyclic groups of orders 2 and 3.
Is there a similar description of the projective special linear ...
- 171
1
vote
0
answers
46
views
Descending FC series
In analogy to the central series one can define a FC series as a sequence $A_i$ of normal subgroups such that
$$
\{1\} = A_0 \lhd A_1 \lhd A_2 \lhd \cdots \lhd A_n = G
$$
such that $A_{i+1}/ A_i$ is ...
- 4,432
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
4
votes
0
answers
124
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Abelian-by-cyclic subgroups of exponential growth solvable groups
I am currently looking for a reference to a proof (or counterexample) to the following statement:
Statement: Assume $G$ is a finitely generated solvable group of exponential growth, then there is a ...
- 4,432
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
5
votes
1
answer
184
views
Explicit short presentation of a 2-generated universal group?
A result of Higman states that there exists a finitely-presented group $G$ in which all other finitely-presented groups embed - I'll call such a group universal. Every countable group embeds in a 2-...
- 5,349
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
17
votes
1
answer
998
views
Where should I search for computations of group cohomology rings of not-too-complicated finite groups?
A computation I'm trying to make uses as input the cohomology rings of not-too-complicated finite groups in low
degrees, and I'd like to determine where to search for preexisting computations.
...
- 6,881
11
votes
0
answers
269
views
Proving a group with two generators is not free that uses the Brahamagupta-Pell equation
Hello I encountered the following while reading a set of notes on free groups. It's not a homework question.
"Does there exist a rational number $\alpha$ with $0 <|\alpha| < 2$ such that the ...
- 111
21
votes
2
answers
2k
views
A new combinatorial property for the character table of a finite group?
Let $G$ be a finite group and $\Lambda = (\lambda_{i,j})$ its character table with $\lambda_{i,1}$ the degree of the ith character.
Consider the following combinatorial property of $\Lambda$: for ...
1
vote
1
answer
194
views
Reference request: The commensurator of an arithmetic lattice is a simple group
I am interested in a reference and proof for some version of the following (folklore?) statement:
``Let $G$ be a (semi)simple Lie group (with no compact factors and trivial centre) and let $\Gamma$ ...
- 855
2
votes
0
answers
143
views
Regular epi- and mono-morphisms for locally compact (Hausdorff) groups
I am interested in what the regular monomorphisms are in the category of locally compact (for me, always Hausdorff) groups (with continuous group homomorphisms).
It is easy to see that the equaliser (...
- 18.7k
1
vote
1
answer
115
views
Bounds for Khukhro-Makarenko theorems
Let’s define the set of outer-commutator group words $OC \subset F_\infty = F[x_0, x_1, …, x_n, …]$ using the following recurrence:
$$\forall i \in \mathbb{N} \text{ } x_i \in OC$$
$$\forall u, v \...
- 2,618
4
votes
0
answers
322
views
Steinberg relations for elementary subgroup of a Chevalley group over an arbitrary ring
Given a semisimple Lie algebra $\frak{g}$ of type $\Phi$ with a Lie algebra representation $\rho:\frak{g}\to \frak{gl}(v)$ and an arbitrary commutative ring one can associate the following gadgets:
...
- 435
0
votes
0
answers
186
views
Subset of reals associated to pairs of matrices in $\mathrm{SL}(2,\mathbb{R})$
Let $\Gamma$ be a subgroup of $\mathrm{SL}(2,\mathbb{R})$. I would like to ask if there is any research on the following set:
$$\Gamma*\Gamma:=\bigg\{\dfrac{(a+b)(a'+b')}{(c+d)(c'+d')}\bigg|\begin{...
- 333
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
4
votes
0
answers
174
views
Algebraic varieties associated to finite groups
Have the following equations been studied in the literature?
Let $G$ be a finite group.
Then I am looking for functions $f : G \rightarrow \mathbb{C}~ \backslash \left\lbrace 0 \right\rbrace $ such ...
- 2,753
5
votes
1
answer
429
views
Cohomology of linear algebraic groups
Let $R$ be a commutative ring. Let $G\subset \mathrm{GL}_m$ be a linear algebraic subgroup. Has the group cohomology $H^i(G(R),R^m)$ been studied in the literature?
For example, do we know
(1) $H^...
user39380
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
3
votes
0
answers
155
views
A variant on the Higman-Thompson groups
Let $C = \mathbb{Z}/d\mathbb{Z}$ ($d \ge 0$).
Let $D = \langle a_c : c \in C, t \mid a^2_c = t^d = 1, ta_ct^{-1} = a_{c+1} \rangle$.
let $E$ be the subgroup generated by $\{a_c : c \in C\}$ and let $...
- 4,728
3
votes
0
answers
165
views
First reference to the Tits alternative
As we know, the "Tits alternative" is a theorem relating to finitely generated linear groups.
I was curious as to where in the literature the Tits alternative is first referred to by this name, as I ...
- 5,146
2
votes
1
answer
82
views
Structure of extensions arising in Lie approximation of connected groups
My imperfect understanding is that, by the work of various authors (Gleason, Yamabe, Montgomery, Zippin ...), the following result is known:
Let $G$ be a connected, locally compact, Hausdorff group, ...
- 25.8k
6
votes
0
answers
164
views
Is a presentation of the hyperbolic orthogonal group of rank 2 over the integers known?
The hyperbolic orthogonal group $O_{g,g}(\mathbb{Z})$ often appears in the study of high-dimensional manifolds, see e.g. work of Kreck or Galatius and Randal-Williams. Let $H$ denote the lattice $\...
- 8,167
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
199
views
Generalization of normal subgroup
I am wondering whether the following concept appears in the group theory literature under some (perhaps different) name. Let $G$ be a group and let $A,B$ be subgroups of $G$.
Definition. Say that $(...
- 1,329
4
votes
1
answer
119
views
Reference for nonquasi-split groups of type $E_6$ and $E_7$ over local fields
The semisimple groups over a local field have been classified by Tits, cf. [1] "Classification of algebraic semisimple groups" in Boulder and [2] "Reductive groups over local fields" in Corvallis.
In ...
- 991
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
9
votes
1
answer
460
views
Connections between linear representations and permutation representations
A finite group $\Gamma$ might be represented by a linear transformation
$$\rho : \Gamma\to\mathrm{GL}(\Bbb R^d),$$
or by permutations
$$\phi :\Gamma\to\mathrm{Sym}(n).$$
Of course, latter ones can ...
- 13.6k
6
votes
1
answer
250
views
Concrete example to illustrate the theory about blocks of groups with cyclic defect groups
I'd like to to have a concrete example to illustrate the theory about blocks of groups with cyclic defect groups.
Thus, I am looking for a finite group $G$ and a prime $p$ dividing $|G|$ satisfying ...
- 1,755
19
votes
0
answers
604
views
How is this group theoretic construct called?
Let $G$ be a finite group, $S\subset G$ a generating set, $|g| = |g|_S = $ word length with respect to $S$. Define the "defect" of $g,h$ to be
$$\psi(g,h) = |g|+|h|-|gh|$$
Then $\psi:G\times G \...
user6671