On transitivity-like properties for torsion-free Abelian groups
-
Andrey R. Chekhlov
and
Patrick W. Keef
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.
Award Identifier / Grant number: 075-02-2022-884
Funding source: Bulgarian National Science Fund
Award Identifier / Grant number: KP-06 No. 32/1
Funding statement: The work of the first-named author is supported by the Ministry of Science and Higher Education of Russia (agreement No. 075-02-2022-884). The work of the second-named author is partially supported by the Bulgarian National Science Fund under Grant KP-06 No. 32/1 of December 07, 2019.
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Articles in the same Issue
- Frontmatter
- Approximating the group algebra of the lamplighter by infinite matrix products
- Shift invariance and reflexivity of compressions of multiplication operators
- Non-invariant deformations of left-invariant complex structures on compact Lie groups
- On the gaps between consecutive primes
- Trivial cup products in bounded cohomology of the free group via aligned chains
- Iwasawa invariants for elliptic curves over ℤp-extensions and Kida's formula
- Estimates of covering type and minimal triangulations based on category weight
- Coherent categorical structures for Lie bialgebras, Manin triples, classical r-matrices and pre-Lie algebras
- Stable vector bundles on generalized Kummer varieties
- Dirac series for complex 𝐸7
- On order of vanishing of characteristic elements
- On transitivity-like properties for torsion-free Abelian groups
- Reverse convolution inequalities for Lebedev–Skalskaya transforms
- Existence of weak solutions to a general class of diffusive shallow medium type equations
Articles in the same Issue
- Frontmatter
- Approximating the group algebra of the lamplighter by infinite matrix products
- Shift invariance and reflexivity of compressions of multiplication operators
- Non-invariant deformations of left-invariant complex structures on compact Lie groups
- On the gaps between consecutive primes
- Trivial cup products in bounded cohomology of the free group via aligned chains
- Iwasawa invariants for elliptic curves over ℤp-extensions and Kida's formula
- Estimates of covering type and minimal triangulations based on category weight
- Coherent categorical structures for Lie bialgebras, Manin triples, classical r-matrices and pre-Lie algebras
- Stable vector bundles on generalized Kummer varieties
- Dirac series for complex 𝐸7
- On order of vanishing of characteristic elements
- On transitivity-like properties for torsion-free Abelian groups
- Reverse convolution inequalities for Lebedev–Skalskaya transforms
- Existence of weak solutions to a general class of diffusive shallow medium type equations
