All Questions
Tagged with reference-request gr.group-theory
700 questions
7
votes
1
answer
476
views
Centralizer of longest element in a finite irreducible Weyl group: related to folding of ADE graphs?
Say $(W,S)$ is a finite Coxeter group, such as a Weyl group (which satisfies an additional crystallographic condition). Assume also that $W$ is irreducible. Then it has a longest element $w_o$ ...
- 52.9k
6
votes
3
answers
965
views
Union of conjugates of a subgroup
Let $G$ be a finite group, $H \leq G$ a proper subgroup. It is well known that the union of the conjugates of $H$ does not cover $G$. I would like to know of more precise results (even in special ...
- 11.3k
2
votes
1
answer
255
views
Does anyone have a copy of Salce's paper "Cotorsion theories for abelian groups"?
The paper "Cotorsion theories for abelian groups" by L. Salce, was published in 1979 in Symposia Math. 21, pages 1-21. According to Google Scholar, it's been cited 233 times, and I keep seeing ...
- 30.3k
3
votes
3
answers
1k
views
A table for irreducible integral representation of finite cyclic groups
Is there such a table where the irreducible integral representations of finite cyclic groups
are listed?
Edited:
Thanks for Todd Leason's comment.Acutally,i want to know all inequivalent ...
- 593
4
votes
0
answers
236
views
Groups inducing edge-colorings on graphs. Is this concept known?
Are the following concepts known in graph/group theory, and if Yes, what are they called and where to read about them? Because I do not know better, I gave them placeholder names for now.
1. ...
- 13.6k
9
votes
1
answer
1k
views
Easy argument for "connected simple real rank zero Lie groups are compact"?
Let $G$ be a connected simple Lie group. It is known that if $G$ has real rank zero, then $G$ is compact.
Background: every connected (semi)simple Lie group $G$ (with Lie algebra $\mathfrak{g}$) has ...
- 223
5
votes
0
answers
228
views
Automorphism groups of cocompact Fuchsian groups as mapping class groups
Let $\Gamma$ be a cocompact Fuchsian group. So it has presentation $$\langle x_1,y_1, \dots, x_g,y_g,z_1, \ldots, z_r \mid [x_1,y_1] \cdots [x_g,y_g]z_1 \cdots z_r=1, \ z_i^{m_i}=1 \rangle$$
for some $...
- 8,401
3
votes
0
answers
217
views
References on a certain generalization of Dedekind groups
Recall that a group is called a Dedekind group if all of its subgroups
are normal. Also recall that a weaker property of a subgroup than normality
is that of being a TI-subgroup: a subgroup $H$ of a ...
- 19.6k
4
votes
1
answer
565
views
Connection between Deligne-Mostow monodromy and Gassner representation at roots of unity of the pure braid group
I am looking for a specific reference to the connection between [1] the Deligne-Mostow monodromy and [2] Gassner representation at roots of unity of the pure braid group. I have seen many references ...
- 11.2k
3
votes
0
answers
183
views
On Khelif's example of a group of countable cofinality having the Bergman property
A group $G$ is defined to have the Bergman property if for any subset $X=X^{-1}$ generating $G$ there exists $n$ such that $X^n=G$.
By a result of Bergman, the permutation group of any set has the ...
- 41.9k
4
votes
0
answers
90
views
Is there a name for this kind of structure? (Not quite a lattice-ordered group)
I'm looking at a certain class of groups $G$ that come with a partial order $\le$ on the elements. So far it looks like $(G,\le)$ has the following properties:
The partial order is invariant under ...
- 4,728
9
votes
0
answers
254
views
Decomposition of linear groups into free products
I recently learnt about Nagao's theorem which states $SL_2(k[t])\cong SL_2(k)\ast_{B(k)}B(k[t])$ for a field $k$. I read in "A.W. Mason, Serre's generalization of Nagao's theorem" that a theorem of ...
- 5,901
19
votes
1
answer
3k
views
On a theorem of Galois
I am currently teaching Galois theory and this week, I mentioned the following theorem of Galois :
Let $P(x) \in \mathbf{Q}[x]$ be an irreducible polynomial of prime degree. Then $P$ is solvable by ...
- 20.8k
4
votes
0
answers
214
views
Hyperbolic subgroups of general linear groups
Is there a classification of hyperbolic subgroups of $GL_n(A)$ for $A$ some ring of characteristic $p$? $A$ for me is a finitely presented algebra over a finite field.
More precisely I'm looking for ...
- 5,901
2
votes
2
answers
594
views
Hausdorff Dimensions of Limit set of subgroups of SL(2,Z)
In a recent paper by Bourgain, Sarnak, Gamburd [1] talks about subgroups of $SL(2,\mathbb{Z})$.
Let $\Lambda$ be a finitely generated non-elementary subgroup of $SL(2,\mathbb{Z})$ with Hausdorff ...
- 22.8k
2
votes
0
answers
126
views
Uncountable chains of countable simple groups -- reference request
Question: Which examples of uncountable ascending / descending chains
of countable simple groups have been described in the literature so far?
Remark: The question is not whether such chains do ...
- 19.6k
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
2
votes
0
answers
87
views
A theory of (or reference for) symmetric point arrangements
I wonder where I can find something written on symmetric point arrangements (see definition below). I am interested in general references, preferably books that introduce (or papers that use) some ...
- 13.6k
6
votes
1
answer
205
views
Understanding (statement of) a theorem of Jack McLaughlin
In the book Twelve sporadic groups, Griess states
If $A$ is an abelian group, $G$ acts on $A$, $z\in Z(G)$ satisfies $z-1\in$ Aut$(G)$, then $H^n(G,A)=0$ for $n\geq 0$. This is an observation of ...
- 1,169
6
votes
1
answer
1k
views
Structure of symplectic group over finite fields
We are working over the finite field $\mathbb{F}_{q}$ of odd prime characteristic $p$ and of cardinality $q$ some power of $p$. We recall the symplectic group $Sp(4,\mathbb{F}_{q})$ as the group of ...
- 85
12
votes
5
answers
797
views
What are the best settings for the large scale geometry of locally compact groups?
My current research involves locally compact groups and from time to time I am tempted to check whether certain notions and statements of geometric group theory of finitely generated groups are still ...
user23860
4
votes
0
answers
218
views
Conjugacy class representatives for the automorphism group of a finite abelian group
Given a finite abelian group $A$, I'd like a list of conjugacy class representatives for its automorphism group ${\rm Aut}(A)$.
In fact, it's not important that I have exactly one representative from ...
- 286
7
votes
0
answers
329
views
A basic question on a base change of a homogeneous space of a linear algebraic group
I asked this basic question in MSE and got a comment "This belongs to Mathoverflow", so I ask my question here.
Let $G$ be a linear algebraic group over a field $k$, and $H\subset G$ be a $k$-...
- 14.2k
3
votes
0
answers
231
views
What is known about "graph algebras"?
In lack for a better name I call a "graph algebra" a simple undirected graph $G=(V,E)$ and a binary mapping $+:E \rightarrow V$ such that:
(1) For all edges $(a,b)$ we have: $a+b \in N(a) \cap N(b)$, ...
user6671
7
votes
1
answer
313
views
Subgroup ranks of the symmetric group
It's well known that every subgroup $G$ of $S_n$ has a generating set of size at most $n-1$ and that this generating set can be found algorithmically (by Jerrum's filter)
I have heard many times a ...
- 539
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
5
votes
3
answers
677
views
Spectrum and scheme of the commutative group-algebra of an abelian group.
The group-algebra of an abelian group is commutative, so we can consider the spectrum of this algebra. Are there any information about the abelian group that we can obtain from such considerations? ...
- 263
2
votes
0
answers
201
views
An example of non trivial projections in a group von Neumann algebra
Let $G$ and $\text{vN}(G)$ be a torsion free group and its group von Neumann algebra. Is there a characterization of non trivial projections in $\text{vN}(G)$? If not, is a certain class of them ...
- 2,118
11
votes
3
answers
3k
views
Reference request for projective representations of finite groups over a non-problematic field
I would like to get a reference where I can learn about the theory of projective representations of finite groups over the complex numbers (or over any field K such that the order of the given group ...
- 2,406
6
votes
2
answers
856
views
Algorithm for Brauer lifting via Brauer tree?
Background: Given a finite group $G$ and a prime $p$ dividing its order, Brauer theory compares the ordinary characters of $G$ with the Brauer characters arising from $p$-modular representations. On ...
- 52.9k
3
votes
1
answer
319
views
Matrix transformation that "rotates" a matrix by $45^\circ$
I have an $n \times n$ integer matrix $A$. I want to obtain an $m \times m$ matrix $B$, where $m \ge n$, such that the rows of $A$ are the diagonals of $B$ and the columns of $A$ are the anti-...
- 2,993
4
votes
2
answers
393
views
Embedding a linearly ordered free monoid into a linearly ordered group
A linearly ordered (shortly, l.o.) monoid is a triple $\mathbb M = (M, \cdot, \le)$ for which $(M, \cdot)$ is a (multiplicatively written) monoid and $\le$ is a total order on $M$ such that $xy < ...
- 10.5k
3
votes
2
answers
1k
views
A structure of the group of automorphisms of an infinite binary tree
My friend asked me to ask his question here. Where he can find (a paper or a book) containing a complete description (with the proof) of a structure of the group of automorphisms of an infinite binary ...
- 5,409
5
votes
2
answers
564
views
Finite groups factorized into two simple alternating groups
My research is somehow related to the following question :
Describe and classify all finite groups $G$ such that $G=HK$ with $H \cap K=1$, where $H \cong A_m$ and $K \cong A_n$ for some integers $m, ...
- 490
2
votes
2
answers
1k
views
Magnus' embedding theorem
I am looking for a (preferably modern) reference to the following old result of Magnus.
Let $F$ be a free group of finite rank and
$$ F_1 = [F,F], F_2 = [F_1,F_1], \dots , F_{n+1} = [F_n,F_n], \dots ...
- 5,050
4
votes
0
answers
83
views
A Krull-Schmidt theorem for partially ordered groups
If $G$ is a po-group (ie. partially ordered group), we say that $G$ is po-indecomposable if it's not the direct product of two non trivial subgroups (such subgroups are necessary convex and normal).
...
- 933
7
votes
1
answer
616
views
Looking for a modern source about Ulm Invariants
I'm looking for a modern, approachable text (preferably a website, textbook, or expository article, and preferably one easily available online or at a library) which can explain the concept of Ulm ...
- 1,979
9
votes
2
answers
674
views
Powers of finite simple groups
I have heard about the following result: for each finite simple non-abelian group $S$ and each natural number $r\ge 2$ there exists a number $n=n(r,S)$ such that the power $S^n$ is $r$-generator but $...
- 331
7
votes
1
answer
839
views
Incomplete Failures of the Inverse Galois Problem
I thought of this question the other day and have not been able to get any traction on references or results along its lines, so I finally caved and decided to ask it here. I am no expert on Galois ...
- 5,191
4
votes
1
answer
353
views
Recognize this countably generated abelian group?
I recently came across a construction that, in abstraction, leads to the following family of abelian groups: Fix $1<q<p$ with $q$ and $p$ relatively prime. The group $G_{(p,q)}$ is given by the ...
- 2,235
9
votes
3
answers
3k
views
Why are divisible abelian groups important?
I just quote wikipedia:
"Divisible groups are important in understanding the structure of abelian groups, especially because they are the injective abelian groups."
I am asking for detail ...
3
votes
2
answers
315
views
Character kernels in the lattice of subgroups of a finite abelian group
I am looking for any efforts that have been made to characterize the character kernels (equivalently, the subgroups yielding cyclic quotients) inside the lattice of subgroups of a finite abelian group....
- 2,851
9
votes
0
answers
230
views
Using Property (T) to approximate invertible matrices
In the wikipedia article for Kazhdan's Property (T), there's an intriguing application:
Similarly, groups with property (T) can be used to construct finite sets of invertible matrices which can ...
3
votes
0
answers
365
views
Coinflation in cohomology
Let $U$ be a normal subgroup of a group $G$ of finite index. On cohomology, somewhat dual to the functorially defined restriction map, $\text{res}^G_U\colon H^n(G, A) \to H^n(U, A)$, the finite index ...
- 423
5
votes
2
answers
332
views
Embedding $G$ in a $Z(G)$ extension of $\operatorname{Aut}G$
This question follows up a question I asked on math.SE. This is a refinement and a reference request.
For what groups $G$ does there exist a $Z(G)$-extension of $\operatorname{Aut}G$ (call it $\tilde ...
- 2,851
3
votes
1
answer
201
views
Injection from Artin monoids to Coxeter groups
Let $\sigma$ be a permutation. If two positive braids represent $\sigma$ and are of minimal length among the braids representing $\sigma$, then they are equal. From what I could gather, this result is ...
- 796
3
votes
1
answer
372
views
Reference for real and complex projective representation of finite group
I'm not a mathematician. I've only learnt about irreducible representation of finite group, symmetric group and simple Lie group. In fact, I don't know projective representation belong to which part ...
- 977
6
votes
1
answer
371
views
Embedding a cancellative monoid into another in such a way that a prescribed element becomes left-invertible
Let $\mathbb A = (A, +_A)$ be a cancellative, but possibly non-commutative, monoid with identity $0$, and fix an element $x \in A$. Does there always exist a cancellative monoid $\mathbb B = (B, +)$ ...
- 10.5k
11
votes
2
answers
854
views
Upper bound on order of finite subgroups of GL_n(Z_p)?
Fix a prime $p$ and integer $n>1$, along with the ring $R$ of integers in a finite extension of the field $\mathbb{Q}_p$ (for example $R = \mathbb{Z}_p$).
Is there an upper bound $C(n,p)$ on ...
- 52.9k
1
vote
1
answer
166
views
Reference for a proof of cancellation property of braid monoids
Let $M$ be a monoid. If $ab=ac$ implies that $b=c$, $a,b,c \in M$, then $M$ is said to have the left cancellation property. Similarly, the right cancellation property is $ba=ca$ implies that $b=c$.
...
- 6,201