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

Some Transitivity-like Concepts in Abelian Groups

The classical notions of transitivity and full transitivity in Abelian p-groups have natural extensions to concepts called Krylov and weak transitivity. The interconnections between these four types of transitivity are determined for Abelian p-groups; there is a marked difference in the relationships when the prime p is equal to 2. In the final section the relationship between full and Krylov transitivity is examined in the case of mixed Abelian groups which are p-local in the sense that multiplication by an integer relatively prime to p is an automorphism

Some transitivity results for torsion Abelian groups

We introduce a new class of fully transitive and transitive Abelian p-groups and study the new concept of weak transitivity which is the missing link between full transitivity and transitivity

Abelian Groups with Partial Decomposition Bases in L\u3csup\u3eδ\u3c/sup\u3e\u3csub\u3e∞ω\u3c/sub\u3e, Part I

We consider the class of abelian groups possessing partial decomposition bases in Lδ∞ω for δ an ordinal. This class contains the class of Warfield groups which are direct summands of simply presented groups or, alternatively, are abelian groups possessing a nice decomposition basis with simply presented cokernel. We prove a classification theorem using numerical invariants that are deduced from the classical Ulm-Kaplansky and Warfield invariants. This extends earlier work by Barwise-Eklof, Göbel and the authors