On Transitivity-like Properties for Torsion-Free Abelian Groups

TL;DR

This paper explores transitivity properties in torsion-free Abelian groups, resolving longstanding problems by constructing examples and extending results from p-torsion cases to the torsion-free setting.

Contribution

It constructs a Krylov transitive torsion-free Abelian group that is neither fully transitive nor transitive, answering a longstanding open problem.

Findings

Constructed a Krylov transitive torsion-free Abelian group not fully transitive.

Extended transitivity results from p-torsion to torsion-free Abelian groups.

Resolved Problem 44 from Krylov-Mikhalev-Tuganbaev's monograph.

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 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).