Universally fully and Krylov transitive torsion-free abelian groups

Extending results from our recent paper in Chekhlov et al. (J Algebra 566(2):187–204, 2021), we define and explore the classes of universally fully transitive and universally Krylov transitive torsion-free Abelian groups. A characterization theorem is proved in which numerous interesting properties of such groups are demonstrated. In addition, we prove the curious fact that these two classes do coincide as well as that in the reduced case these groups are just homogeneous separable and thus, in particular, they are both fully transitive and transitive. Some related results pertaining to H-full transitivity and H-Krylov transitivity for some special (fixed) groups H which, in particular, can be viewed as subgroups of a torsion-free Abelian group G are also obtained. Our achieved here results somewhat strengthen those established by Goldsmith and Strungmann (Commun Algebra 33(4):1177–1191, 2005).