Related papers: Associativity of the Commutator Operation in Group…

A group is irreducibly represented if it has a faithful irreducible unitary representation. For countable groups, a criterion for irreducible representability is given, which generalises a result obtained for finite groups by W. Gasch\"utz…

We analzye Rieffel's construction of generalized fixed point algebras in the setting of group actions on Hilbert modules. Let G be a locally compact group acting on a C*-algebra B. We construct a Hilbert module F over the reduced crossed…

The following discourse is inspired by the works on hyperbolic groups of Epstein, and Neumann/Reeves. Epstein showed that geometrically finite hyperbolic groups are biautomatic. Neumann/Reeves showed that virtually central extensions of…

Given a finite group $G$ and a commutative ring $G$-spectrum $R$, we study the separable commutative algebras in the category of compact $R$-modules. We isolate three conditions on the geometric fixed points of $R$ which ensure that every…

Two finite groups are said to have the same order type if for each positive integer $n$ both groups have the same number of elements of order $n$. In 1987 John G. Thompson asked if in this case the solvability of one group implies the…

Let $G$ be a finite group. For $x \in G$, we define the solvabilizer of $x$ in $G$, denoted $sol_G(x)$, to be the set $\{g \in G \mid \langle g,x \rangle$ is solvable$\}$. A group $G$ is an S-group if $sol_G(x)$ is a subgroup of $G$ for…

For every countable abelian group $G$ we find the set of all its subgroups $H$ ($H\leq G$) such that a typical measure-preserving $H$-action on a standard atomless probability space $(X,\mathcal{F}, \mu)$ can be extended to a free…

We prove that Thompson's groups $F$ and $V$ are the geometry groups of associativity, and of associativity together with commutativity, respectively. We deduce new presentations of $F$ and $V$. These presentations lead to considering a…

Let $G$ be a finite group and $\sigma_1(G)=\frac{1}{|G|}\sum_{H\leq G}\,|H|$. In this note, we prove that if $\sigma_1(G)<\frac{117}{20}$, then $G$ is solvable. Moreover, we have $\sigma_1(G)=\frac{117}{20}$ if and only if $G\cong A_5$.…

Thompson's groups, which are denoted by $F, T$ and $V$, were introduced by R. Thompson. It is known that they are related to various fields in mathematics. In this paper, we establish that Thompson's groups are regarded as subgroups of…

Let $G$ be a profinite group. We prove that the commutator subgroup $G'$ is finite-by-procyclic if and only if the set of all commutators of $G$ is contained in a union of countably many procyclic subgroups.

We consider a combinatorial problem occurring naturally in a group theoretical setting and provide a constructive solution in a special case. More precisely, in 1999 the author established a logarithmic bound for the derived length of the…

A double magma is a nonempty set with two binary operations satisfying the interchange law. We call a double magma proper if the two operations are distinct and commutative if the operations are commutative. A double semigroup is a double…

We introduce and investigate different definitions of effective amenability, in terms of computability of F{\o}lner sets, Reiter functions, and F{\o}lner functions. As a consequence, we prove that recursively presented amenable groups have…

We consider a field $F$ and positive integers $n$, $m$, such that $m$ is not divisible by $\mathrm{Char}(F)$ and is prime to $n!$. The absolute Galois group $G_F$ acts on the group $\mathbb{U}_n(\mathbb{Z}/m)$ of all $(n+1)\times(n+1)$…

In this second part we prove that, if $G$ is one of the groups $\mathrm{PSL}_2(q)$ with $q>5$ and $q\equiv 5\pmod {24}$ or $q\equiv 13 \pmod{24}$, then the fundamental group of every acyclic $2$-dimensional, fixed point free and finite…

This paper is devoted to the study of noncommutative ergodic theorems for connected amenable locally compact groups. For a dynamical system $(\mathcal{M},\tau,G,\sigma)$, where $(\mathcal{M},\tau)$ is a von Neumann algebra with a normal…

Shumyatsky and the second author proved that if G is a finitely generated residually finite p-group satisfying a law, then, for almost all primes, the fact that a normal and commutator-closed set of generators satisfies a positive law…

A group G is almost cyclic if there is an element x in G, such that for all g in G, there is an element y in G and an integer n with ygy^{-1} = x^n (that is, every element is conjugate to some power of x). W. Ziller asked whether there are…

Let X be a set and Bin(X) the set of all binary operations on X. We say that a subset of Bin(X) is a distributive set of operations if all pairs of elements are right distributive. J.Przytycki posed the question of which groups can be…