On transitivity-like properties for torsion-free Abelian groups

Abstract We study some close relationships between the classes of transitive, fully transitive and Krylov torsion-free Abelian groups. In addition, as an application achieved assertions, we resolve old-standing problems, posed by Krylov, Mikhalev Tuganbaev in their monograph [P. A. V. Tuganbaev, Endomorphism Rings Groups, Kluwer Academic, Dordrecht, 2003]. Specifically, answer Problem 44 from there affirmative constructing a group which is neither nor transitive. This extends to case certain similar results p -torsion case. We, alternatively, also expand version concerning transitivity, full transitivity -primary