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The unit version of the Kaplansky conjecture is about units in $FG$ where $F$ is a field and $G$ is a torsion free group. In a recent counter example by Giles Gardam, it is given an e …

Let $G$ be a topological group. We define an equivalence relation on $G$ as follows: For $a,b\in G$ we set $a\sim b$ if the following two maps are topologically conjugate: $$x\mapsto ax,\qquad x\maps …

Assume that a functor on the category of groups vanishes on all projective objects. Is it necessarily the left derived functor of a half exact functor on this category?

Is there a discrete non-abelian group whose dual in a reasonable sense is isomorphic to the solenoid constructed via a sequence of quaternions $S^3$ instead of a sequence of circles? The motivation c …

What is an example of a torsion free discrete abelian group $G$ whose dual space $\hat{G}$ is not a path connected space? The Motivation: The motivation comes from the idempotent conjecture of Kapla …

Let $G$ be a compact topological group with normalized Haar measure $\mu$. Is there an effective isometric action of $G$ on some $\mathbb{R}^n$ such that the following map would be a non-irreducible …

Let $A$ be a unital $C^*$-algebra which is equipped with a faithful trace $T$. In particular we may consider $A=C^*_{\text{red}} (G)$ for some discrete group $G$. We consider the following differentia …

$\DeclareMathOperator\Cov{Cov}$Motivation: If $G$ is a finite group and $\phi=X+iY: G\to \mathbb{T}$ is a character of $G$, then $\Cov(X,Y)=0$ where $X$, $Y$ are considered as two real random variab …

On the objects of the category of groups we define the mapping $G\mapsto \operatorname{Hol}(G)$, the holomorph $G\rtimes \operatorname{Aut}(G)$ of $G$. Can we extend this mapping to a functor on this …

Let $\mathcal{FG}$ be the category of finite groups. Let $S$ be a full subcategory of $\mathcal{FG}$. Assume that $G\in \mathcal{FG}$ and $P\in S$ is a subgroup of $G$. We say that $P$ is $S$-maximal …

Let $k\in \mathbb{N}$ be a natural number and $p$ be a prime number with $p\nmid k$. (We thank Prof. Bartel for his comment on the latter non divisibility condition.) We denote by $U(p)=\{1,2,\ldots …

1)Let $G$ be a countable discrete group. Can $G$ be embbeded in a locally connected Lie group? 2)let $G$ be a countable discrete group with a prescribed generating set and corresponding word metr …

Motivation: It is obvious that for a finite index subgroup $H$ of a group $G$, there exists a normal subgroup $K$ of $G$, $K\subset H$, with $|G/K|<\infty$. Our question: Let $G$ be a topological gro …

Revision: According to comment of Wojowu we give a complete revise for this post. A group $G$ is a pr-group if all projections of $C^*_{\text{red}} G$ are contained in its dense subalgebra $\mathbb{ …

Edit: According to comment conversations we revise the question. Let $G$ be a group. We consider the following subset of $G$: $$\{g\in G \mid e^{\lambda_g} \in \mathbb{C}\lambda (G)\},$$ where $\lamb …