Related papers: The Tarski numbers of groups

Let G be an abelian group and let lambda be the smallest rank of any group whose direct sum with a free group is isomorphic to G. If lambda is uncountable, then G has lambda pairwise disjoint, non-free subgroups. There is an example where…

Let $K/F$ be a finite Galois extension of number fields with Galois group $G$, let $A$ be an abelian variety defined over $F$, and let ${\cyr W}(A_{^{/ K}})$ and ${\cyr W}(A_{^{/ F}})$ denote, respectively, the Tate-Shafarevich groups of…

We study the number of elements $x$ and $y$ of a finite group $G$ such that $x \otimes y= 1_{_{G \otimes G}}$ in the nonabelian tensor square $G \otimes G$ of $G$. This number, divided by $|G|^2$, is called the tensor degree of $G$ and has…

We study the solvable groups $G$ that have an irreducible character $\chi\in \Irr(G)$ such that $\chi \bar{\chi}$ has at most two non-principal irreducible constituents.

A non-trivial element of a group is a generalized torsion element if some products of its conjugates is the identity. The minimum number of such conjugates is called a generalized torsion order. We provide several restrictions for…

We investigate topologies on groups which arise naturally from their algebraic structure, including the Frech\'et-Markov, Hausdorff-Markov, and various kinds of Zariski topologies. Answering a question by Dikranjan and Toller, we show that…

We show that (with one possible exception) there exist strongly dense free subgroups in any semisimple algebraic group over a large enough field. These are nonabelian free subgroups all of whose subgroups are either cyclic or Zariski dense.…

It is proven that if $G$ is a finite group, then $G^\omega$ has $2^{\mathfrak c}$ dense nonmeasurable subgroups. Also, other examples of compact groups with dense nonmeasurable subgroups are presented.

We construct first examples of infinite finitely generated residually finite torsion groups with positive rank gradient. In particular, these groups are non-amenable. Some applications to problems about cost and $L^2$-Betti numbers are…

Tkachenko and Yaschenko [34] characterized the abelian groups G such that all proper unconditionally closed subsets of G are finite, these are precisely the abelian groups G having cofinite Zariski topology (they proved that such a G is…

A numerical semigroup is an additive subsemigroup of the natural numbers that contains zero and has finite complement. A numerical semigroup is irreducible if it cannot be written as an intersection of numerical semigroups properly…

There has been some interest on how the average character degree affects the structure of a finite group. We define, and denote by $ \mathrm{anz}(G) $, the average number of zeros of characters of a finite group $ G $ as the number of zeros…

We give a complete classification of finitely generated virtually free groups up to $\forall\exists$-elementary equivalence. As a corollary, we give an algorithm that takes as input two finite presentations of virtually free groups, and…

The article contains a survey of our results on weakly commensurable arithmetic and general Zariski-dense subgroups, length-commensurable and isospectral locally symmetric spaces and of related problems in the theory of semi-simple agebraic…

The purpose of this article is to present a survey of our recent results on length commensurable and isospectral locally symmetric spaces. The geometric questions led us to the notion of "weak commensurability" of two Zariski-dense…

A group $G$ is invariably generated if there exists a subset $S \subseteq G$ such that, for every choice $g_s \in G$ for $s \in S$, the group $G$ is generated by $\{ s^{g_s} \mid s \in S \}$. In [GGJ16] Gelander, Golan and Juschenko showed…

Let $G$ be a multiplicatively written finite group. We denote by $\mathsf E(G)$ the smallest integer $t$ such that every sequence of $t$ elements in $G$ contains a product-one subsequence of length $|G|$. In 1961, Erd\H{o}s, Ginzburg and…

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…

It is well known that if $G$ is a countable amenable group and $G \curvearrowright (Y, \nu)$ factors onto $G \curvearrowright (X, \mu)$, then the entropy of the first action must be greater than or equal to the entropy of the second action.…

Let $\chi$ be an irreducible character of a group $G,$ and $S_c(G)=\sum_{\chi\in {\rm Irr}(G)}{\rm cod}(\chi)$ be the sum of the codegrees of the irreducible characters of $G.$ Write ${\rm fcod} (G)=\frac{S_c(G)}{|G|}.$ We aim to explore…