All Questions
Tagged with reference-request gr.group-theory
700 questions
14
votes
0
answers
552
views
Who conjectured that a transitive projective plane is Desarguesian?
The only known finite projective plane with a transitive automorphism group is the Desarguesian plane $PG(2,q)$ and it seems likely that there are no others, although this is not (quite) proved.
...
- 12.7k
4
votes
2
answers
998
views
A construction of generators of discrete subgroups of SL(2,R)
I know about geometrical method of construction of discrete subgroups of $SL(2,\mathbb{R})$ using Lobachevsky plane (e.g. B.A. Dubrovin, A.T. Fomenko, S.P. Novikov, Modern Geometry --- Methods and ...
- 211
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
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
13
votes
4
answers
1k
views
Simple groups with the same cardinality as PSL_2(Z/p)
In an undergrad honors algebra course it's sometimes shown that when $p$ is prime and $>3$ then
$PSL_2(Z/p)$ is simple of of order $p(p-1)(p+1)/2$. But that this is the "only" simple group
having ...
11
votes
1
answer
676
views
Analysis and finitely generated groups
Dear all, this is perhaps a bit a vague question, but some references would already be very helpfull.
So let $G$ be a finitely generated group and choose some finite set of generators. This allows to ...
- 8,098
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
5
votes
1
answer
264
views
Group not leaving subset invariant
Let $Y,X$ be two sets of size n,m. Let $Y\subset X$.
What is the maximal group(in size) $G< Sym(X)$ such that gY=Y imply that $g=1$?
Here I mean that the only permutation which permutes elements of ...
- 2,179
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
2
votes
1
answer
274
views
virtual chain conditions in groups
In group theory, it's often very useful to know whether a family of subgroups (eg normal subgroups, Zariski-closed subgroups, ...) satisfies an ascending chain condition or a descending chain ...
- 4,728
7
votes
1
answer
958
views
Groups whose normal subgroups form a chain with respect to inclusion
Let G be a finite group. In general, given two normal subgroups N and K of G, we need not have N < K or K < N. The easiest example is the Klein 4-group V4 and its subgroups of order 2. So assume ...
- 307
5
votes
2
answers
984
views
Automorphism Group of some Classical groups
Hi All,
I would like to know the Automorphism group of some simple classical groups, such as PSL(n,q) or some PSU or PSp groups. Could you please give me some recommended books or papers then? I ...
- 233
8
votes
4
answers
1k
views
Degree of commutativity of finite groups and subgroups
Recently I started reading some articles about the
degree of commutativity of finite groups. I have some questions:
In "Subgroup commutativity degrees of finite groups" Tarnauceanu
proposes ...
user11178
13
votes
1
answer
736
views
Idempotent measures on the free binary system?
Let $(S,*)$ be the free (non associative) binary system on one generator (so $S$ is just the set of terms in $*$ and $1$). There is an extension of $*$ to the space $P(S)$ of finitely additive ...
- 3,547
7
votes
1
answer
818
views
Uncertainty principle for non-commutative groups
Is it true that for every group $G$ and $f\in \mathbb C[G]$ it holds that $$\dim(\mathbb C[G]*f)\mathop{supp}(f)\geq |G| ?$$
Here, $\mathbb C[G]$ is the group algebra, and by $\mathbb C[G]*f$ I ...
- 2,179
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
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
9
votes
1
answer
1k
views
Ping Pong and Free Group Factors
This question concerns alternative characterizations of free group factors. The ping pong lemma is a well-known criteria for the freeness of a group. I've often wondered if there is a ping pong like ...
- 7,067
10
votes
4
answers
1k
views
Twist of a group Hopf-algebra
Let $G$ be a finite group with identity element $e$, and $C[G]$ the ring of complex-valued functions on $G$, with pointwise addition and multiplication. Then $C[G]$ is naturally a Hopf algebra, with ...
- 13.3k
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
4
votes
1
answer
328
views
Algebraic K-groups and braids
This is (I think) a reference request:
Are there calculations of any algebraic K-groups for the (group ring of) the Artin braid groups?
- 1,180
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
3
votes
1
answer
696
views
Unique factorization of finite groups under direct sum?
I am told that finite groups have unique factorization under direct product. That is, call a nontrivial group "indivisible" if it is not isomorphic to a direct product of nontrivial groups. Then ...
- 4,994
8
votes
0
answers
252
views
Amenability versus the ideal of wandering sets
Let $G$ be a finitely generated group acting on a set $S$ (on the right). Define the heirarchy of "marginal sets" as follows:
The emptyset is 0-marginal.
A set E is $(k+1)$-marginal if $E$ can be ...
- 3,547
12
votes
2
answers
770
views
Is "subamenable" the same as amenable?
Let $G$ be a finitely generated group. Does the following condition imply the amenability of $G$:
there is a function $\mu:\mathcal{P}(G) \to [0,1]$ such that:
(subadditive) $\mu(G) = 1$, $\mu(A \...
- 3,547
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 ...
11
votes
1
answer
3k
views
Where can I easily look up / calculate (abelian) group cohomology?
For an example I'm trying to understand, I need to calculate some cohomology group of some $\mathbb Z$-module with coefficients in some other $\mathbb Z$-module (with no interesting actions). (In ...
- 54.6k
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
5
votes
5
answers
873
views
Green polynomials
Is there any software for calculating Green polynomials (of type A)? Or, at least, where can I find tables of Green polynomials? Also, I would be interested in some formulas for Green polynomials in ...
- 1,590
4
votes
2
answers
1k
views
Reference request for two-generator subgroups of a free group
According to B. Fine, G. Rosenberger, On restricted Gromov groups, Comm. Algebra 20 (1992) 2171--2181, Gromov proved the following in his long article introducing word-hyperbolic groups:
Let $x$ ...
- 25.8k
2
votes
1
answer
185
views
Kurosh radical theory for topological groups?
Does anyone know if there has been much work done on radical and semisimple classes in the sense of Kurosh within the category of topological groups (or subcategories thereof)? For instance, for a ...
- 4,728
2
votes
3
answers
2k
views
Infinite Field Theory and Category Theory
I should start by saying that I have not studied field theory in depth, so if this question is totally off base, I apologize. Something I noticed as I studied group theory is many concepts that were ...
- 5,759
5
votes
0
answers
219
views
Character tables of the p-core of the binary modular congruence group of p-power level
Let $p \geq 5$ be a prime and let $n$ be positive integer. In his Ph.D thesis (See The characters of binary modular congruence group, Bulletin of the
American Mathematical Society. 79 (1973), no. 4.), ...
- 2,396
2
votes
2
answers
1k
views
description of functions of conditionally negative type on a group
Recall that a kernel conditionaly of negative type on a set $X$ is a map $\psi:X\times X\rightarrow\mathbb{R}$ with the following properties:
1) $\psi(x,x)=0$
2) $\psi(y,x)=\psi(x,y)$
3) for any ...
- 1,222
6
votes
1
answer
900
views
Reconstruction Conjecture: Group theoretic formulation
As we read from wiki, informally, the reconstruction conjecture in graph theory says that graphs are determined uniquely by their subgraphs.
Is there a group-theoretic formulation of this conjecture?
...
- 2,855
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 ...
10
votes
2
answers
3k
views
Doubly-transitive groups
I want to know what all doubly-transitive groups look like. Do you know some good reference where I can read about it?
- 2,179
4
votes
1
answer
371
views
Normal subgroups of binary polyhedral groups (reference request)
The binary polyhedral groups are finite subgroups of the quaternions corresponding (via McKay's ADE classification) to the $E$ series of affine Dynkin diagrams. They are also the lifts to $\mathrm{...
- 33.3k
8
votes
2
answers
3k
views
Centralizers in GL(n,p)
There appear to be a number of rational canonical forms. The best thing about standards is how many there are to choose from. However, the standard I choose seems to have a centralizer that is ...
- 10.7k
7
votes
2
answers
529
views
Telling group algebras apart
It's a big, famous, hard problem in operator algebras to determine if the von Neumann algebras $L(F_2)$ and $L(F_3)$ are isomorphic, or not. Here $F_n$ is the free group on n generators and $L(F_n)$ ...
- 18.7k
10
votes
3
answers
1k
views
Random walks and Lyapunov exponents
Given a sequence $Y_1, Y_2, \dots$ of i.i.d. matrices in $\mathrm{GL}_n(\mathbb R)$, there is a theorem of Furstenberg and Kesten which says that if $\mathbb E(\log\|Y_1\|)$ is finite, there exists a ...
- 315
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
41
votes
8
answers
17k
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 ...
7
votes
2
answers
780
views
Finite groups with a character having very few nonzero values?
A number theorist I know (who studies Galois representations) raised a question recently:
Which finite groups can have an irreducible character of degree at least 2 having only $n=2, 3$, or 4 ...
- 52.9k
5
votes
2
answers
346
views
Reference request: A theorem by S. Garrison
A theorem by S. Garrison states that if $G$ is a finite solvable group and $|cd(G)| = 4$ then $dl(G)\leq |cd(G)|$ (the Taketa inequality, which is conjectured to hold for all finite solvable groups). ...
- 2,508
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
44
votes
10
answers
11k
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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
2
votes
1
answer
226
views
Name of the Marshall Hall paper in which he proved that the intersection of all subgroups of a fixed finite index is again finite index?
can someone please tell me? I couldn't find a reference in the paper I was reading.
- 87
17
votes
5
answers
3k
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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
52
votes
14
answers
14k
views
Introductory text on geometric group theory?
Can someone indicate me a good introductory text on geometric group theory?