Torsion-Free Weakly Transitive Abelian Groups

We introduce the notion of weak transitivity for torsion-free abelian groups. A torsion-free abelian group G is called weakly transitive if for any pair of elements x, y ∈ G and endomorphisms ϕ, ψ ∈ End(G) such that xϕ = y, yψ = x, there exists an automorphism of G mapping x onto y. It is shown that every suitable ring can be realized as the endomorphism ring of a weakly transitive torsion-free abelian group, and we characterize up to a number-theoretical property the separable weakly transitive torsion-free abelian groups

Relationships between Almost Completely Decomposable Abelian Groups with Their Multiplication Groups

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 $G$ is an Abelian group with respect to addition. An Abelian group $G$ with multiplication, defined on it, is called a \textsf{ring on the group} $G$. Let $\mathcal{A}_0$ be the class of Abelian block-rigid almost completely decomposable groups of ring type with cyclic regulator quotient. In the paper, we study relationships between the above groups and their multiplication groups. It is proved that groups from $\mathcal{A}_0$ are definable by their multiplication groups. For a rigid group $G\in\mathcal{A}_0$, the isomorphism problem is solved: we describe multiplications from $\text{Mult}\,G$ that define isomorphic rings on $G$. We describe Abelian groups that are realized as the multiplication group of some group in $\mathcal{A}_0$. We also describe groups in $\mathcal{A}_0$ that are isomorphic to their multiplication groups.Comment: arXiv admin note: text overlap with arXiv:2205.1065

Torsion-Free Groups and Modules with the Involution Property

An Abelian group or module is said to have the involution property if every endomorphism is the sum of two automorphisms, one of which is an involution. We investigate this property for completely decomposable torsion-free Abelian groups and modules over the ring of -adic integers

Butler groups of infinite rank

AbstractButler groups are torsion-free abelian groups which — in the infinite rank case — can be defined in two different ways. One definition requires that all the balanced extensions of torsion groups by them are splitting, while the other stipulates that they admit continuous transfinite chains (with finite rank factors) of so-called decent subgroups.This paper is devoted to the three major questions for Butler groups of infinite rank: Are the two definitions equivalent? Are balanced subgroups of completely decomposable torsion-free groups always Butler groups? Which pure subgroups of Butler groups are again Butler groups? In attacking these problems, a new approach is used by utilizing ℵ0-prebalanced chains and relative balanced-projective resolutions introduced by Bican and Fuchs [5].A noteworthy feature is that no additional set-theoretical hypotheses are needed

Torsion-free Abelian Groups Revisited

Let G be a torsion--free abelian group of finite rank. The automorphism group Aut(G) acts on the set of maximal independent subsets of G. The orbits of this action are the isomorphism classes of indecomposable decompositions of G. G contains a direct sum of strongly indecomposable groups as a characteristic subgroup of finite index, giving rise to a classification of finite rank strongly indecomposable torsion--free abelian groups.Comment: In Section 5, The quasi--category of V, I erroneously concluded that the unit group of the ring $Q\otimes End(G)$ is $Q^*.Aut(G)$. This dumb error led to several false statements, so I wish to withdraw and re-write the whole pape