11
votes
2answers
358 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
0
votes
1answer
192 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 ...
3
votes
1answer
145 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 ...
4
votes
0answers
165 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 ...
13
votes
2answers
771 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 $G$ ...
2
votes
2answers
597 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
2
votes
2answers
403 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 ...
0
votes
1answer
405 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 ...
21
votes
10answers
3k 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
1answer
769 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. ...
2
votes
5answers
1k 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 ...
4
votes
2answers
241 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 ...
2
votes
1answer
237 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 ...