Related papers: Associativity of the Commutator Operation in Group…

Let $G$ be a finite group and $\sigma_1(G)=\frac{1}{|G|}\sum_{H\leq G}\,|H|$. In this paper, we prove that if $\sigma_1(G)<2+\frac{11}{|G|}$\,, then $G$ is supersolvable. In particular, some new characterizations of the well-known groups…

An infinite family of association schemes obtained from the general unitary groups acting transitively on the sets of isotropic vectors in the finite unitary spaces are investigated. We compute the parameters and determine the character…

This paper generalizes the basic notions of additive and multiplicative combinatorics to the setting of group actions: if $G$ is a group acting on a set $X$, and we have subsets $A\subseteq G$ and $Y\subseteq X$ such that the set of pairs…

An element $x$ of a group $G$ is a commutator if it can be expressed in the form $x = a^{-1}b^{-1}ab$ for some $a, b \in G$. In 2010 MacHale posed the following problem in the Kourovka notebook: does there exist a finite group $G$, with…

We establish an uncountable amenable ergodic Roth theorem, in which the acting group is not assumed to be countable and the space need not be separable. This generalizes a previous result of Bergelson, McCutcheon and Zhang, and complements…

Guba and Sapir asked if the simultaneous conjugacy problem was solvable in Diagram Groups or, at least, for Thompson's group F. We give a solution to the latter question using elementary techniques which rely purely on the description of F…

In this work we investigate tensor completions of groups by associative rings, which were introduced by R.Lyndon and G.Baumslag in 1960s. The main result states that there exists an algorithm that decides if a given finite system of…

Bogopolski, Martino and Ventura in [BMV10] introduced a general criteria to construct groups extensions with unsolvable conjugacy problem using short exact sequences. We prove that such extensions have always solvable word problem. This…

A binary mean operation m(x,y) is said to be compatible with a semigroup law *, if * satisfies the Gauss' functional equation m(x,y) * m(x,y) = x * y for all x, y. Thus the arithmetic mean is compatible with the group addition in the set of…

We give a new formulation of some of our recent results on the following problem: if all uniformly bounded representations on a discrete group $G$ are similar to unitary ones, is the group amenable? In \S 5, we give a new proof of…

William W. Boone and Graham Higman proved that a finitely generated group has soluble word problem if and only if it can be embedded in a simple group that can be embedded in a finitely presented group. We prove the exact analogue for…

Every finite group $G$ has a normal series each of whose factors is either a solvable group or a direct product of non-abelian simple groups. The minimum number of nonsolvable factors, attained on all possible such series in $G$, is called…

In 1972, B. E. Johnson proved that a locally compact group $G$ is amenable if and only if certain Hochschild cohomology groups of its convolution algebra $L^1(G)$ vanish. Similarly, $G$ is compact if and only if $L^1(G)$ is biprojective: In…

We study and extend the conditions for asociativity on fusion over antielement free $\sigma$-sets to introduce a group to solve $\sigma$-set equations. $\sigma$-sets as a theory of sets and antisets is sumarized and used as a framework to…

For every Turing machine, we construct an automaton group that simulates it. Precisely, starting from an initial configuration of the Turing machine, we explicitly construct an element of the group such that the Turing machine stops if, and…

We prove a structure theorem for stable functions on amenable groups, which extends the arithmetic regularity lemma for stable subsets of finite groups. Given a group $G$, a function $f\colon G\to [-1,1]$ is called stable if the binary…

Consider $ A^* $, the free monoid generated by the finite alphabet $A$ with the concatenation operation. Two words have the same commutative image when one is a permutation of the symbols of the other. The commutative closure of a set $ L…

Let $R$ be a subset of a group $G$. We call a subgroup $H$ of $G$ the $R$-conjugate-permutable subgroup of $G$, if $HH^{x}=H^{x}H$ for all $x\in R$. This concept is a generalization of conjugate-permutable subgroups introduced by T. Foguel.…

In this paper, we consider some generalized commutator equations in a finite group and show that the number of solutions of such equations are characters of that group. We also obtain explicit formula for this character, considering the…

We study the bounded cohomology of certain groups acting on the Cantor set. More specifically, we consider the full group of homeomorphisms of the Cantor set as well as Thompson's group $V$. We prove that both of these groups are boundedly…