On transitivity-like properties for torsion-free Abelian groups

Abstract

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