All Questions
Tagged with examples gr.group-theory
36
questions
9
votes
2
answers
630
views
Torsion-free virtually free-by-cyclic groups
Is it known if there are any examples of a finitely generated group $G$ such that:
$G$ has a finite index subgroup $H$ which is free-by-cyclic
$G$ itself is not free-by-cyclic
$G$ is torsion-free
...
- 283
2
votes
1
answer
117
views
Example of maximal multicurve complex
in this paper we have :
" On the Teichmüller tower of mapping class groups By Allen Hatcher at Ithaca, Pierre Lochak at Paris and Leila Schneps."
Definition. The maximal multicurve complex $...
- 119
0
votes
0
answers
99
views
On simple examples of unimodularity
$w=z=x+ 1 =y−1$ provides $wz−xy=w^2−(w−1)(w+ 1) = 1$. Hence if $x,y$ are odd then $w,z$ are even and all four integers are close.
Is there elementary example where only $w$ is even and all four ...
- 13.5k
0
votes
1
answer
112
views
Examples of non-proper profinite HNN extensions
We define a profinite HNN extension as the profinite completion of the abstract HNN extension. In the abstract case, the homomorphim of the base group to the HNN extension is always a monomorphism. ...
3
votes
1
answer
93
views
On cospectral graphs
Is there examples of non-isomorphic cospectral graphs having
Non-isomorphic automorphism groups?
Isomorphic automorphism groups?
- 13.5k
0
votes
1
answer
192
views
about simple non-abelian 2-generated group [closed]
Does there exist a simple non-abelian 2-generated group $G$ and two elements $a, b \in G$, such that $\langle \{a, b\} \rangle = G$, $a^2 =1$ and $\forall c, d \in G$ $\langle \{c^{-1}bc, d^{-1}bd \} \...
- 184
20
votes
0
answers
562
views
Infinitely generated non-free group with all proper subgroups free
Is there any example of group $G$ satisfying the following properties?
$G$ is non-abelian, infinitely generated (i.e. it is not finitely generated) and not a free group.
$H< G$ implies that $H$ is ...
- 599
15
votes
1
answer
1k
views
Examples of hyperbolic groups
What are some other classes of word-hyperbolic groups other than the finite groups, fundamental groups of surfaces with Euler characteristics negative and virtually free groups?
- 305
18
votes
7
answers
2k
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 ...
10
votes
1
answer
225
views
Naturally occurring, non-amenable Zappa-Szep products of discrete amenable groups?
We say $G$ is the Zappa-Szep product of two subgroups $K$ and $P$ if $K\cap P = \{e\}$ and the function $K\times P \to G$, $(k,p)\mapsto kp$, is bijective.
The Iwasawa decomposition shows that we can ...
- 25.3k
2
votes
0
answers
150
views
An example of a finite group with some specific permutable subgroups
The following question is about finite groups.
Let $G$ be a finite group and let $H, K \leqslant G$. We say that $H$ permutes with $K$ if $HK = KH$ and in this case $HK \leqslant G$.
The Symbol $\pi ...
- 141
24
votes
8
answers
3k
views
Applications of logic to group theory?
There seems to be an ever-growing literature on the first-order theory of groups. While I find this interaction between group theory and logic quite appealing, I was wondering the following:
Are ...
- 359
15
votes
1
answer
520
views
Torsion-free abelian group $A$ such that $A \not \simeq A \oplus \Bbb Z \simeq A \oplus \Bbb Z^2$
Is there a torsion-free abelian group $A$ such that $A \not \simeq A \oplus \Bbb Z \simeq A \oplus \Bbb Z \oplus \Bbb Z$ (as groups)?
Notice that $\Bbb Z$ is not cancellable, so
$A \oplus \Bbb Z \...
- 1,682
3
votes
1
answer
168
views
p-Group satisfying the minimal condition on abelian subgroups
Are there examples of $p$-groups satisfying the minimal condition on abelian subgroups but do not satisfying the minimal condition on subgroups?
Obviously such a group cannot be locally finite.
I've ...
- 137
2
votes
0
answers
147
views
Actions on spaces with measured walls
In geometric group theory, the question of whether or not a group acts nicely on a CAT(0) cube complex, or equivalently on a median graph, is of interest. The same question for actions on spaces with ...
- 2,361
91
votes
2
answers
7k
views
$A$ is isomorphic to $A \oplus \mathbb{Z}^2$, but not to $A \oplus \mathbb{Z}$
Are there abelian groups $A$ with $A \cong A \oplus \mathbb{Z}^2$ and $A \not\cong A \oplus \mathbb{Z}$?
8
votes
2
answers
263
views
Roller's problem on median groups
At the end of his dissertation Poc Sets, Median Algebras and Group Actions, Martin Roller asks
A group $G$ is called median if it acts freely and transitively on a median algebra. This is ...
- 2,361
7
votes
0
answers
376
views
Linear vs smooth actions of finite groups on spheres, euclidean spaces and closed disks
I would like to know examples (with references, if possible) of the following:
(1) a finite group $G$ acting effectively and smoothly on a sphere $S^n$ (any $n$) but admitting no effective linear ...
6
votes
0
answers
395
views
Example of a torsionfree group satisfying a cohomological condition
Let us call a finitely generated group $G$ cohomologically rich if for each $k \geq 0$, we can find a subgroup $G'$ and a prime $p$ such that $H^k(G';\mathbb F_p) \neq 0$. Examples which come to mind ...
- 11.8k
6
votes
0
answers
406
views
An example of a simple infinite 2-group
I've asked this question before on Mathematics, and they suggested me to ask here (Link).
Is there an example of a simple infinite $2$-group?
Informations
If a $2$-group is Artinian I know that it ...
- 599
0
votes
1
answer
256
views
Example of a polycyclic group which is not of polynomial growth? [closed]
The title already says everything: What is an example of a polycyclic group $G$ which is not of polynomial growth (equivalently, by Gromov's theorem, which is not virtually nilpotent)?
- 11.8k
11
votes
2
answers
466
views
A sequence of subsets of an infinite group
Is there an infinite group $G$ such that there is not any sequence $(A_n)$ of its subsets such that always
$$A_n=A_n^{-1}, \quad A_{n+1}A_{n+1}\subsetneqq A_n$$
?
link
- 261
0
votes
1
answer
351
views
Examples of groups such that order isomorphism of the subgroups of $G\times G$ and $H\times H$ does not imply isomorphism of $G$ and $H$
Let $G$ and $H$ be groups, $\operatorname{Sub}(G\times G)$ be the set of all subgroups of $G\times G$ and $\operatorname{Sub}(H\times H)$ be the set of all subgroups of $H\times H$. Assume there ...
- 1,145
4
votes
1
answer
211
views
A group 3-cocycle, trivial on a pair of generating subgroups?
I'm looking for an example of the following situation:
A group $G$ generated by finite subgroups $H$ and $K$,
a non-trivial 3-cocycle $\omega \in H^3(G, \mathbb{k}^\times)$
such that
the ...
- 7,490
4
votes
0
answers
270
views
Interesting commensurated subgroups of countable groups
Let $G$ be a group and let $K$ be a subgroup. Say $K$ is commensurated in $G$ if $gKg^{-1} \cap K$ has finite index in $K$ for all $g \in G$. Commensurated subgroups are an inherent feature of ...
- 4,678
13
votes
2
answers
1k
views
Does such an infinite index subgroup exist?
Notation: If $G$ is a countable group and $H$ is a subgroup, for $g\in G$, let $|\mathcal{O}_{gH}|$ be the size of the $H$-orbit of $gH$ in the $H$-set $G/H$.
Does there exist a countable group $G$ ...
- 5,192
2
votes
2
answers
846
views
Non-split groups
I am looking for a reference with definitions on what it means for an algebraic group to be split, quasi-split, and non-split. I would like to see some examples of the different "types".
Thanks,
Tom
24
votes
7
answers
3k
views
Which group does not satisfy the Tits alternative?
A group is said to satisfy the Tits alternative if every finitely generated subgroup of $G$ is either virtually solvable or contains a nonabelian free subgroup.
Tits proved this for linear groups, ...
- 4,198
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
0
votes
1
answer
476
views
Must finite groups with isomorphic commutators and quotients be isomorphic?
Let G and H be finite groups. Let G' = [G,G] and H' = [H,H] be the corresponding derived groups (commutator subgroups) of G and H. I am looking for an example where G' is isomorphic to H' and G/G' is ...
- 677
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, ...
0
votes
1
answer
1k
views
Maximal subgroups of abelian groups and Q-algebras
Let $G$ be an abelian group which does not have a maximal subgroup. Does it follow that $G$ is a $\mathbb{Q}$-algebra?
It is easy to see that $\mathbb{Q}$-algebras do not admit any maximal subgroups. ...
3
votes
5
answers
2k
views
non-abelian groups of prescribed order
Is there a construction that will give a non-abelian group of order $p^mr$ where $p$ is a prime, $r$ and $p$ are relatively prime and $m$ is an arbitrary non-negative integer? I suspect in this ...
user577
4
votes
2
answers
319
views
Is there a poset with 0 with countable automorphism group?
Is there a poset P with a unique least element, such that every element is covered by finitely many other elements of P (and P is locally finite -- actually, per David Speyer's example, let's say that ...
- 12.5k
2
votes
1
answer
262
views
Hausdorff Derived Series
There is a short section in the book Locally Compact Groups by Markus Stroppel (Chapter B7) on the notion of a "Hausdorff Solvable Group", which he defines as a topological group with a descending ...
- 33
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]."
...