All Questions
Tagged with reference-request gr.group-theory
700 questions
73
votes
9
answers
9k
views
What are "classical groups"?
Unlike many other terms in mathematics which have a universally understood meaning (for instance, "group"), the term classical group seems to have a fuzzier definition. Apparently it originates with ...
- 52.9k
62
votes
9
answers
9k
views
Fundamental groups of noncompact surfaces
I got fantastic answers to my previous question (about modern references for the fact that surfaces can be triangulated), so I thought I'd ask a related question. A basic fact about surface topology ...
- 44.8k
52
votes
14
answers
14k
views
Introductory text on geometric group theory?
Can someone indicate me a good introductory text on geometric group theory?
44
votes
10
answers
11k
views
The finite subgroups of SL(2,C)
Books can be written about the finite subgroups of $\mathrm{SL}(2,\mathbb C)$ (and their immediate family, like the polyhedral groups...) I am about to start writing notes for a short course about ...
- 47.3k
42
votes
6
answers
4k
views
Measures of non-abelian-ness
Let $G$ be a finite non-abelian group of $n$ elements.
I would like a measure that intuitively captures the
extent to which $G$ is non-commutative.
One easy measure is a count of the non-commutative ...
- 151k
41
votes
8
answers
16k
views
What are some good group theory references?
I'm curious about what books people use for a group theory reference. I don't currently own a dedicated group theory book, and I think I'd find such a book very helpful in my research. What is your ...
35
votes
7
answers
4k
views
Paradoxical Mathematical Objects Pending for Construction [duplicate]
The possible properties and applications of some mathematical objects have been described far before their rigorous mathematical definition. Some of them even had a seemingly paradoxical description ...
35
votes
6
answers
5k
views
Character-free proof that Frobenius kernel is a normal subgroup?
The question is in the title, but here is some background/reminders:
A subgroup $H\neq\{1\}$ of a finite group $G$ is called a Frobenius complement if $H\cap H^g = \{1\}$ for all $g\in G\backslash H$....
- 13k
32
votes
0
answers
993
views
Is there a Mathieu groupoid M_31?
I have read something which said that the large amount of common structure between the simple groups $SL(3,3)$ and $M_{11}$ indicated to Conway the possibility that the Mathieu groupoid $M_{13}$ might ...
- 3,645
30
votes
1
answer
2k
views
How strong is this conjecture? $(Z/nZ)^*$ is generated by "small" elements
Conjecture: There are constants $c,k$ such that every $(Z/nZ)^*$ is generated by its elements smaller than $k (\log n)^c$.
Where $(Z/nZ)^*$ is the multiplicative group of integers mod $n$. My main ...
- 4,529
30
votes
1
answer
592
views
Guess that group via product queries
Suppose someone (person B) knows a finite group $G$ of order $n$.
You (person A) know only the order $n$,
and that $1$ is the name of the identity element.
The group elements are named $1,2,\ldots,n$ ...
- 151k
30
votes
0
answers
999
views
Follow-up to Steinberg's problem (12) in his 1966 ICM talk?
Steinberg's lecture at the 1966 ICM in Moscow here surveyed his work on regular elements of semisimple algebraic groups, while also formulating a number of then-open questions as "problems" (...
- 52.9k
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
27
votes
1
answer
3k
views
An anecdote by R. Schmidt
Did anybody here ever read those lines by R. Schmidt (?) where he talked about the terseness of articles in group theory in the days prior to the conclusion of the classification of the finite simple ...
27
votes
1
answer
1k
views
Properties to have matrices that commute in $\mathrm{GL}_n(\mathbb C)$
Let $G$ be a finite subgroup of $\mathrm{GL}_n(\mathbb C)$, $A,B \in G$ whose eigenvalues are thus in the unit circle.
Assume that the eigenvalues of $A$ are included in a circle arc of length $<\...
user49093
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
24
votes
3
answers
3k
views
Non-abelian Grothendieck group
By general nonsense the forgetful functor from groups to monoids has a left adjoint. It maps a monoid $(X,\cdot,1)$ to the free group on $\{\underline{x} : x \in X\}$ modulo the relations $\underline{...
- 63.1k
24
votes
2
answers
1k
views
Why can the general quintic be transformed to $v^5-5\beta v^3+10\beta^2v-\beta^2 = 0$?
The quintic can be transformed to the one-parameter Brioschi quintic,
$$u^5-10\alpha u^3+45\alpha^2u-\alpha^2 = 0\tag1$$
This form is well-known for its connection to the symmetries of the ...
- 12.6k
24
votes
2
answers
3k
views
Does any textbook take this approach to the isomorphism theorems?
Below, I present an outline of a proof of the first isomorphism theorem for groups. This is how I usually think of the first isomorphism theorem for ______________, but groups will get the points ...
- 12.1k
23
votes
2
answers
2k
views
Modern references on hyperbolic groups
Several good references dedicated to hyperbolic groups have been written until 1990, including:
Hyperbolic groups, written by M. Gromov.
Géométrie et théorie des groupes : les groupes hyperboliques ...
- 8,401
22
votes
2
answers
2k
views
Proofs of the Stallings-Swan theorem
It is a well-known and deep${}^\ast$ theorem that if a group $G$ has cohomological dimension one then it must be free. This was proved in the late 60's by Stallings (for finitely generated groups) and ...
- 35.9k
22
votes
2
answers
2k
views
Roadmap to learning the classification of finite simple groups
I want to learn the classification of finite simple groups. But it is often commented that it is a theorem spanning tens of thousands of pages of research papers. So it is quite intimidating to an ...
- 820
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 ...
21
votes
1
answer
564
views
Partitions of ${\rm Sym}(\mathbb{N})$ induced by convergent, but not absolutely convergent series
Let $(a_n) \subset \mathbb{R}$ be a sequence such that the series
$\sum_{n=1}^\infty a_n$ converges, but does not converge absolutely.
Then there is a partition of the symmetric group ${\rm Sym}(\...
- 19.6k
19
votes
2
answers
1k
views
Does the amenability problem for Thompson's group $F$ predate 1980?
The first place where the amenability problem for Thompson's group $F$ appears in the literature is, I believe, 1980 in a problems article by Ross Geoghegan. I have heard, however, vague comments to ...
- 3,547
19
votes
2
answers
943
views
Reference for the triple covering of A_6
I would like to ask for a reference (book, paper ...) for the following nice construction, which I have found as an exercise in some notes of a course by R. Borcherds. For $n=6$ or $7$ (and only in ...
- 38k
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
19
votes
2
answers
1k
views
Reference request for Plancherel measure
I need a good reference for the basic definitions of the dual of locally compact group (not necessarily abelian), its natural topology, $\sigma$-algebra, and the Plancherel measure on it (when they ...
- 26k
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
18
votes
3
answers
745
views
Number of primitive $n$th roots with positive versus negative real parts
Does anyone know a reference to the following results, which I can prove, but I suspect may be known. Let $R(n)$ denote the number of primitive $n$th roots of unity with positive real part, and $L(n)$ ...
- 1,991
18
votes
2
answers
1k
views
Has anyone seen this construction of the Weil representation of $\mathrm{Sp}_{2k}(\mathbb{F}_p)$?
$\def\FF{\mathbb{F}}\def\CC{\mathbb{C}}\def\QQ{\mathbb{Q}}\def\Sp{\text{Sp}}\def\SL{\text{SL}}\def\GL{\text{GL}}\def\PGL{\text{PGL}}$Let $p$ be an odd prime. The Weil representation is a $p^k$-...
- 156k
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
17
votes
3
answers
1k
views
Examples of locally hyperbolic groups
It is well-known that a subgroup of a hyperbolic group need not be hyperbolic. Let us say that a (finitely generated) group $G$ is locally hyperbolic if all its finitely generated subgroups are (...
- 383
17
votes
5
answers
3k
views
Reference for this theorem in representation theory?
Let $G$ be a finite group and $\chi$ be an irreducible character of
$G$ (characteristic zero algebraically closed base field). If $H$ is
the kernel of $\chi$ then the irreducible representations of $G/...
- 887
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
17
votes
1
answer
2k
views
A synopsis of Adyan’s solution to the general Burnside problem?
Where can I find a high-level overview
of Adyan’s original proof of the existence of finitely generated infinite groups with finite exponent?
Additionally:
If possible, would an expert please ...
- 7,067
17
votes
1
answer
1k
views
Explicit character tables of non-existent finite simple groups
In connection with the historical development of the classification of finite simple groups, I am interested in a particular aspect that seems to be less well-documented than the main narrative of ...
17
votes
3
answers
815
views
Does this subgroup of "even braids" have a name?
The full braid group on $n$ strands $B_n$ admits a surjective homomorphism $p\colon\thinspace B_n\to \Sigma_n$ onto the symmetric group on $n$ letters, which takes a braid to the induced permutation ...
- 35.9k
17
votes
1
answer
459
views
Existence of a quasi-isometric residually finite group?
It's, by now, more or less well known that residual finiteness is not a quasi-isometry invariant for finitely generated groups (see here for an example). Thus the following question makes sense:
...
- 733
17
votes
3
answers
1k
views
How to find more (finite almost simple) groups with a given Sylow subgroup
I'm looking for some examples of actions on Sylow p-groups, and often those actions appear in the case of finite almost simple groups. Given a finite almost simple group, I understand in principle ...
- 10.7k
17
votes
1
answer
575
views
Group cochains invariant under the action of the symmetric group
Let $G$ be a finite group and $A$ an abelian group. Recall the cochain groups
$$
C^k = \{f: G^k \to A\}
$$
and the coboundary map
$$
\delta : C^k \to C^{k+1}
$$
$$
(\delta f)(g_1, \ldots, ...
- 12.8k
16
votes
3
answers
2k
views
What are the main open problems in the theory of amenability of groups?
I have read the Paterson and Runde books about amenability of groups, but I do not know what are the most intriguing questions in this area today.
A survey or a list of questions would be welcome.
16
votes
2
answers
818
views
Decomposing $(\mathbb C^n)^{\otimes m}$ as a representation of $S_n\times S_m$
$V=\mathbb C^n$ is a $\mathbb CS_n$-module, where $S_n$ is the symmetric group of degree $n$, via the representation sending a permutation to the corresponding permutation matrix. The tensor power $V^...
- 38.6k
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
16
votes
3
answers
1k
views
Torsion subgroups of hyperbolic groups are finite?
Is it true that torsion subgroups of hyperbolic groups are finite?
I have a vague memory that this is true, perhaps due to Ol'shanskii, but have been struggling to find a reference.
(By a torsion ...
- 161
16
votes
3
answers
1k
views
Conjectures in the representation theory of the symmetric group
Question: What are current open conjectures about the representation theory of the symmetric group?
I am interested mostly in the characteristic 0 case, but conjectures for the modular case can also ...
- 26.5k
16
votes
2
answers
992
views
Maximal number of maximal subgroups
Let $G$ be a finite group. I want to find an upper bound on the number of the maximal subgroups. My questions is does it possible to prove that the number of maximal subgroups of any finite group $G$ ...
- 2,179
16
votes
1
answer
408
views
Is there a relationship between Broué's abelian defect group conjecture and Alperin's weight conjecture?
Let $G$ be a finite group, let $k$ be a large enough field of characteristic $p>0$. Let $p\mid |G|$.
Broué's abelian defect group conjecture states the following:
Let $B$ be a block of $kG$ with
...
- 1,755
16
votes
3
answers
1k
views
What are some interesting examples of non-classical dynamical systems? (Group action other than $\mathbb{Z}$ or $\mathbb{R}$ )
By classical dynamical system, I mean a measure space together with a measurable action of the integers or the reals. Of course, this action is often interpreted as evolution with respect to discrete ...
16
votes
3
answers
2k
views
Your favorite papers on geometric group theory
I would like to improve my "depth of understanding" in geometric group theory. So I am interested in short and accessible papers on subjects related to this field but which are not always ...