All Questions
Tagged with big-list gr.group-theory
35 questions
3
votes
1
answer
263
views
Nonisomorphic central products on the same pair of groups?
A central product of two groups $G$ and $H$ is determined as follows. The groups $G$ and $H$ have respective central subgroups $A$ and $B$ which are isomorphic, let $\delta:A\rightarrow B$ be such ...
6
votes
0
answers
244
views
What can lattices tell us about lattices?
A general group-theoretic lattice is usually defined as something like
A discrete subgroup $\Gamma$ of a locally compact group $G$ is a lattice if the quotient $G/\Gamma$ carries a $G$-invariant ...
- 2,183
71
votes
28
answers
8k
views
Results from abstract algebra which look wrong (but are true)
There are many statements in abstract algebra, often asked by beginners, which are just too good to be true. For example, if $N$ is a normal subgroup of a group $G$, is $G/N$ isomorphic to a subgroup ...
6
votes
2
answers
351
views
Differences between $p$-groups and $q$-groups
First, let me include the same disclaimer that goes in the first line of any article I write: all groups considered herein are finite.
Academically, I work with connecting the arithmetic structure of ...
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.
9
votes
7
answers
765
views
Important combinatorial and algebraic interpretations of the coefficients in the polynomial $[n]!_q = (1+q)(1+q+q^2) \ldots (1+q+\cdots + q^{n-1})$
What are some important combinatorial and algebraic interpretations of the coefficients in the polynomial
$$[n]!_q = (1+q)(1+q+q^2) \ldots (1+q+\cdots + q^{n-1})?$$
As motivation, I will give ...
4
votes
1
answer
416
views
Functors on the category of abelian groups which satisfy $F(G\times H) \cong F(G)\otimes_{\mathbb{Z}} F(H)$
Edit: According to the comment of Todd Trimble, I revise the question.
What are some examples of functors $F$ on the category of Abelian groups or category of rings which satisfy $$F(G\times H)\cong ...
1
vote
1
answer
312
views
Topological Invariants for Group
Let $\mathbf{Grp}$ be the category of groups and $\mathbf{Top}$ be the category of topological spaces. To each group $(G, \circ_G)$, we can associate a topological space $(G,\tau_G)$ the basis for ...
user57432
4
votes
0
answers
270
views
Has the external knit product been used to construct a previously unknown group?
In the Wikipedia article
Zappa–Szép product
, the knit product (a.k.a. Zappa–Szép product, Zappa–Rédei-Szép product, general product, exact factorization) is defined, and its basic properties are laid ...
10
votes
1
answer
2k
views
Open problems concerning all the finite groups
What are the open problems concerning all the finite groups?
The references will be appreciated. Here are two examples:
Aschbacher-Guralnick conjecture (AG1984 p.447): the number of conjugacy ...
3
votes
3
answers
569
views
Free ergodic probability measure-preserving actions of the free group
Let $(X,\mathcal{B},\mu)$ be a standard Borel probability space. Let $\Gamma$ be a countable group.
An action of $\Gamma$ on $X$ is:
essentially free if for all $g \in \Gamma \setminus \{e \}$,...
9
votes
0
answers
182
views
$p$-groups and the arithmetic of $p$
I cannot think of an example of a property of a $p$-group or a pro-$p$ group that depends on the arithmetic of the prime number $p$. To clarify l do not mean the size of $p$. Clearly lots of stuff ...
- 5,450
18
votes
7
answers
3k
views
Examples of residually-finite groups
One of the main reasons I only supervised one PhD student is that I find it hard to find an appropriate topic for a PhD project. A good approach, in my view, is to have on the one hand a list of ...
8
votes
2
answers
850
views
Geometric or topological results from group theory
Do you know interesting examples of purely geometric or topological results which can be proved using group theory? To make precise what I have in mind, let us consider the two following examples:
...
9
votes
1
answer
753
views
List of finitely presented groups with undecidable word problem
Is there any reasonably updated list of (representative) examples of finitely presented groups with undecidable word problem?
By "representative" I mean "avoiding obvious redundancy", i.e. examples ...
- 343
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 ...
85
votes
23
answers
11k
views
Solving algebraic problems with topology
Often, topologists reduce a problem which is - in some sense - of geometric nature, into an algebraic question that is then (partiallly) solved to give back some understanding of the original problem.
...
0
votes
0
answers
75
views
The set of (property) elements of a locally compact group is closed
For which properties $(P)$ is the following statement known to be true?
In any locally compact group $G$, the elements of $G$ that satisfy $(P)$ form a closed subset of $G$. In other words, the ...
- 4,728
56
votes
14
answers
21k
views
Fantastic properties of Z/2Z
Recently I gave a lecture to master's students about some nice properties of the group with two elements $\mathbb{Z}/2\mathbb{Z}$. Typically, I wanted to present simple, natural situations where the ...
4
votes
1
answer
556
views
Symmetries of the standard probability space
The standard probability space $(I, \mathcal B, \lambda)$ consists of the interval $I = [0,1]$, its Borel $\sigma$-algebra $\mathcal B := \mathcal B(I)$ and Lebesgue measure $\lambda$. In applications,...
16
votes
3
answers
1k
views
How can I tell if a group is linear?
The basic question is in the title, but I am interested in both necessary and sufficient conditions.
I know the Tits' alternative and Malcev's result that finitely generated linear groups are ...
9
votes
3
answers
735
views
Judging whether a finitely presented group is a 3-manifold group?
Given a finitely presented group $G$, how many necessary conditions do people know for $G$ to be isomorphic to the fundamental group of some closed connected 3-manifold? (e.g. residually finite)
8
votes
0
answers
1k
views
Examples of uncountable abelian $p$-groups
Does anyone know of any interesting examples of an infinite abelian $p$-group which is uncountable?
By non-interesting here I mean the direct sums of cylic and quasi-cylic groups, and totally ...
6
votes
4
answers
647
views
Examples of results first proved using geometrical methods?
Hi all,
I am beginning to learn about geometric group theory. I would like to write a little exposé intended to be read by the uninitated, so it would be nice to talk about (preferably simple) ...
2
votes
9
answers
2k
views
Examples of amenable groups other than finite groups
I'm reading about amenable groups. What are explicit examples of nonabelian discrete amenable groups other than finite groups? Perhaps a group presentation or matrix representation would be useful.
38
votes
7
answers
4k
views
Lie group examples
I'm looking for interesting applications of Lie groups for an introductory Lie groups graduate course. In particular I'd like to hear of non-standard examples that at first sight do not seem to be ...
41
votes
3
answers
3k
views
Names of finite groups
Question: If you have a finite group, how do you name it?
If, for whatever reason, you have to list all subgroups of $GL_2({\mathbb F}_5)$ up to isomorphism in a paper, you are likely to write ...
2
votes
2
answers
528
views
Characterizations of amenable groups which use the space $\ell_1(G)$ and convolution
Let $G$ be a discrete group.
Do you know characterizations of amenable groups which use the space $\ell_1(G)$ and convolution?
I only know Johnson's theorem:
A group is amenable if and only if the ...
40
votes
6
answers
5k
views
What are some interesting corollaries of the classification of finite simple groups?
The classification of finite simple groups, whether it be viewed as finished, or as a work in progress, is (or will be) without doubt an enormous achievement. It clearly sheds a great deal of light on ...
8
votes
3
answers
9k
views
Applications of Group Theory Which Motivate Theoretical Questions?
I'm going to be a teaching assistant for an undergraduate class in abstract algebra next semester, for students who have not taken abstract algebra before. It will deal with group theory and linear ...
25
votes
10
answers
4k
views
Examples of non-abelian groups arising in nature without any natural action
It's said that most groups arise through their actions. For instance, Galois groups arise in Galois theory as automorphisms of field extensions. Linear groups arise as automorphisms of vector spaces, ...
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 ...
9
votes
7
answers
3k
views
Hopfian and Co-Hopfian groups (examples)
Hi,
I'm looking for examples of groups that are both Hopfian and Co-Hopfian. I have a non trivial (and beautiful, at least to me) example: $\mathrm{SL}(n,\mathbb{Z})$ (with $n>2$).
Do you know ...
12
votes
3
answers
2k
views
Database of finite presentations of used groups
Do You know any kind of database of presentations of groups?
It may be on-line or off-line in form of tables, ideally case would be integrated in some Computer Algebra System. I am interested the ...
36
votes
17
answers
6k
views
Canonical examples of algebraic structures
Please list some examples of common examples of algebraic structures. I was thinking answers of the following form.
"When I read about a [insert structure here], I immediately think of [example]."
...