Related papers: Abelian supplements in almost simple groups

In this paper, we study precompact abelian groups G that contain no sequence {x_n} such that {0} \cup {\pm x_n : n \in N} is infinite and quasi-convex in G, and x_n --> 0. We characterize groups with this property in the following classes…

It is proved that every group of the form $G=AB$ with two subgroups $A$ and $B$ each of which is either abelian or has a quasicyclic subgroup of index $2$ is soluble of derived length at most $3$. In particular, if $A$ is abelian and $B$ is…

The subgroup of the Schur multiplier of a finite group G consisting of all cohomology classes whose restriction to any abelian subgroup of G is zero is called the Bogomolov multiplier of G. We prove that if G is quasisimple or almost…

We generalize two of our previous results on abelian definable groups in $p$-adically closed fields to the non-abelian case. First, we show that if $G$ is a definable group that is not definably compact, then $G$ has a one-dimensional…

We study the subgroup B_0(G) of H^2(G,Q/Z) consisting of all elements which have trivial restrictions to every Abelian subgroup of G. The group B_0(G) serves as the simplest nontrivial obstruction to stable rationality of algebraic…

We say that a subgroup $H$ is isolated in a group $G$ if for every $x\in G$ we have either $x\in H$ or $\langle x\rangle\cap H=1$. In this short note, we describe the set of isolated subgroups of a finite abelian group. The technique used…

For an Abelian group $G$, any homomorphism $\mu\colon G\otimes G\rightarrow G$ is called a \textsf{multiplication} on $G$. The set $\text{Mult}\,G$ of all multiplications on an Abelian group $G$ itself is an Abelian group with respect to…

A group is small if it has countably many complete $n$-types over the empty set for each natural number n. More generally, a group $G$ is weakly small if it has countably many complete 1-types over every finite subset of G. We show here…

We study notions such as finite presentability and coherence, for partially ordered abelian groups and vector spaces. Typical results are the following: (i) A partially ordered abelian group G is finitely presented if and only…

Given a group $G$ with bounded torsion that acts properly on a systolic complex, we show that every solvable subgroup of $G$ is finitely generated and virtually abelian of rank at most $2$. In particular this gives a new proof of the above…

If a soluble group $G$ contains two finitely generated abelian subgroups $A,B$ such that the number of double cosets $AgB$ is finite, then $G$ is shown to be virtually polycyclic.

This is an expository work presenting in detail the proof of the structure theorem for divisible abelian groups. A divisible abelian group is an abelian group that satisfies nD=D for all natural n. The theorem states that any divisible…

Given a subset $W$ of an abelian group $G$, a subset $C$ is called an additive complement for $W$ if $W+C=G$; if, moreover, no proper subset of $C$ has this property, then we say that $C$ is a minimal complement for $W$. It is natural to…

Let $G$ be a finite abelian group and $s$ be a positive integer. A subset $A$ of $G$ is called a {\em perfect $s$-basis of $G$} if each element of $G$ can be written uniquely as the sum of at most $s$ (not-necessarily-distinct) elements of…

The authors investigate the structure of quasi-o-minimal groups. Among other results, they show that quasi-o-minimal groups are abelian, that quasi-o-minimal densely ordered archimedian groups are divisible, and that every divisible…

Let G be a finite abelian group and F a field such that char(F) does not divide |G|. Denote by FG the group algebra of G over F. A (semisimple) abelian code is an ideal of FG. Two codes I and J of FG are G-equivalent if there exists an…

In this paper, we classify the finite simple groups with an abelian Sylow subgroup.

This paper targets to generalize the notion of Hopfian groups in the commutative case by defining the so-called {\bf relatively Hopfian groups} and {\bf weakly Hopfian groups}, and establishing some their crucial properties and…

We say that finite groups are isospectral if they have the same sets of orders of elements. It is known that every nonsolvable finite group $G$ isospectral to a finite simple group has a unique nonabelian composition factor, that is, the…

Let $\mathcal{C}$ be a class of finite groups closed for subgroups, quotients groups and extensions. Let $\Gamma$ be a finite simplicial graph and $G = G_{\Gamma}$ be the corresponding pro-$\mathcal C$ RAAG. We show that if $N$ is a…