  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 oldstanding
problems, posed by Krylov-Mikhalev-Tuganbaev in their monograph [17].
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, obtained by Braun et al. in J. Algebra
(2019). 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 due Files-Goldsmith from Proc. Amer. Math. Soc. (1998) and
Danchev-Goldsmith from J. Comm. Algebra (2011).
