‎given an integer $n$‎, ‎we denote by $mathfrak b_n$ and $mathfrak c_n$ the classes of all groups $g$ for which the map $phi_{n}:gmapsto g^n$ is a monomorphism and an epimorphism of $g$‎, ‎respectively‎. ‎in this paper we give a characterization for groups in $mathfrak b_n$ and for groups in $mathfrak c_n$‎. ‎we also obtain an arithmetic description of the set of all integers $n$ such that a gr...

‎the number of factorizations of a finite abelian group as the product of two subgroups is computed in two different ways and a combinatorial identity involving gaussian binomial coefficients is presented‎.

we prove that the class of profinite groups $g$ that have a factorization $g=ab$‎ ‎with $a$ and $b$ abelian closed subgroups‎, ‎is closed under taking inverse limits‎ ‎of surjective inverse systems‎. ‎this is a generalization of a recent result by k. h. hofmann and f. g. russo‎. ‎as an application we reprove their generalization of iwasawa's structure theorem for‎ ‎quasihamiltonian pro-$p$...

We deal with some pcf, (posible cofinality theory) investigations mostly motivated by questions in abelian group theory. We concentrate on applications to test problems but we expect the combinatorics will have reasonably wide applications. We almost always answer the test problem which is proving the existence of ℵω-free abelian groups with trivial dual, i.e., with no non-trivial homomorphisms...

Here are three recently-established theorems from the literature. (A) (2006) Every non-metrizable compact abelian group K has 2|K|-many proper dense pseudocompact subgroups. (B) (2003) Every non-metrizable compact abelian group K admits 22 |K| -many strictly finer pseudocompact topological group refinements. (C) (2007) Every non-metrizable pseudocompact abelian group has a proper dense pseudoco...

We examine the existence of universal elements in classes of infinite abelian groups. The main method is using group invariants which are defined relative to club guessing sequences. We prove, for example: Theorem: For n ≥ 2, there is a purely universal separable p-group in אn if, and only if, 20 ≤ אn. §0 Introduction In this paper “group” will always mean “infinite abelian group”, and “cardina...

We study effective categoricity of computable abelian groups of the form ⊕ i∈ω H, where H is a subgroup of (Q,+). Such groups are called homogeneous completely decomposable. It is well-known that a homogeneous completely decomposable group is computably categorical if and only if its rank is finite. We study ∆n-categoricity in this class of groups, for n > 1. We introduce a new algebraic concep...