Related papers: Solution to the Uniformly Fully Inert Subgroups Pr…

In \cite{branko} Gr{\"u}nbaum asked if the set of all affine invariant points of a given convex body is equal to the set of all points invariant under every affine automorphism of the body. In \cite{ivan} we have proven the case of a body…

We give a reduction to quasisimple groups for Donovan's conjecture for blocks with abelian defect groups defined with respect to a suitable discrete valuation ring $\mathcal{O}$. Consequences are that Donovan's conjecture holds for…

We prove that the isomorphism problem is decidable for generalized Baumslag-Solitar (GBS) groups with one quasi-conjugacy class and full support gaps. In order to do so we introduce a family of invariants that fully characterize the…

The purpose of the article is to provide an unified way to formulate zero-sum invariants. Let $G$ be a finite additive abelian group. Let $B(G)$ denote the set consisting of all nonempty zero-sum sequences over G. For $\Omega \subset B(G$),…

Consider an expansion $\mathcal R=(R,<,+,\ldots)$ of an ordered divisible Abelian group of finite burden defining no nonempty subset $X$ of $R$ which is dense and codense in a definable open subset $U$ of $R$ with $X \subseteq U$. We…

It is proved that any infinite Abelian group of infinite exponent admits a non-discrete reflexive group topology.

More than 50 years ago, Laszlo Fuchs asked which abelian groups can be the group of units of a commutative ring. Though progress has been made, the question remains open. We provide an answer to this question in the case of indecomposable…

Relying on the techniques and ideas from our recent paper [13], we prove several anti-classification results for various rigidity conditions in countable abelian and nilpotent groups. We prove three main theorems: (1) the rigid abelian…

For Abelian p-groups, Goldsmith, Salce, et al., introduced the notion of minimal full inertia. In parallel to this, we define the concept of minimal characteristic inertia and explore those p-primary Abelian groups having minimal…

Zassenhaus Conjecture for torsion units states that every augmentation one torsion unit of the integral group ring of a finite group G is conjugate to an element of G in the units of rational group algebra QG. This conjecture has been…

A subset $D$ of an Abelian group is $decomposable$ if $\emptyset\ne D\subset D+D$. In the paper we give partial answer to an open problem asking whether every finite decomposable subset $D$ of an Abelian group contains a non-empty subset…

Let $G$ be a finite abelian group. We show that its Davenport constant $D(G)$ satisfies $D(G)\leq \exp(G)+\frac{|G|}{\exp(G)}-1$, provided that $\exp(G)\geq\sqrt{|G|}$, and $D(G)\leq 2\sqrt{|G|}-1$, if $\exp(G)<\sqrt{|G|}$. This proves a…

In a recent paper, Gabriel Navarro and Pham Huu Tiep show that the so-called Alperin Weight Conjecture can be verified via the Classification of the Finite Simple Groups, provided any simple group fulfills a very precise list of conditions.…

We show that it is consistent with ZFC that every compact group has a non-Haar-measurable subgroup. In addition, we demonstrate a natural construction, and we conjecture that this construction always produces a non-measurable subgroup of a…

For a partition $\underline{\lambda} = (\lambda_{1}^{\rho _1}>\lambda_{2}^{\rho _2}>\lambda_{3}^{\rho _3}>\ldots>\lambda_{k}^{\rho _k})$ and its associated finite $\mathcal{R}$-module…

We generalize the positive solution of the Frobenius conjecture and refinements thereof by studying the structure of groups that admit a fix-point-free automorphism satisfying an identity. We show, in particular, that for every polynomial…

In their study of the Yamabe problem in the presence of isometry group, Hebey and Vaugon announced a conjecture. This conjecture generalizes Aubin's conjecture, which has already been proven and is sufficient to solve the Yamabe problem. In…

We give a solution stated in the title to problem 3 of part 1 of the problems listed in the book of Eklof and Mekler [EM],(p.453). There, in pp. 241-242, this is discussed and proved in some cases. The existence of strongly lambda-free ones…

We give a necessary and sufficient condition, in terms of a certain reflection principle, for every unconditionally closed subset of a group G to be algebraic. As a corollary, we prove that this is always the case when G is a direct product…

This paper provides a complete classification of the subvarieties and subquasivarieties of pointed Abelian lattice-ordered groups ($\ell$-groups) that are generated by their totally ordered members. We present two complementary approaches…