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Universally fully and Krylov transitive torsion-free abelian groups

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Abstract

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 Strüngmann (Commun Algebra 33(4):1177–1191, 2005).

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Acknowledgements

The second named author P.V. Danchev is thankful to Professor Adolf Mader from Hawaii, Honolulu, United States, for their private communication concerning the present topic. The all three authors of this article are also very appreciated to the referee for the careful reading of the text and given expert comments and suggestions in his/her report as well as to the handling editor, Professor John S. Wilson from Oxford, United Kingdom, for his professional editorial management.

Funding

The work of the second named author P.V. Danchev is partially supported by the Bulgarian National Science Fund under Grant KP-06 No. 32/1 of December 07, 2019.

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Authors and Affiliations

  1. Department of Mathematics and Mechanics, Tomsk State University, Tomsk, Russia, 634050

    Andrey R. Chekhlov

  2. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113, Sofia, Bulgaria

    Peter V. Danchev

  3. Department of Mathematics, Whitman College, Walla Walla, WA, 99362, USA

    Patrick W. Keef

Authors
  1. Andrey R. Chekhlov
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  2. Peter V. Danchev
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  3. Patrick W. Keef
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Correspondence to Peter V. Danchev.

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Communicated by John S. Wilson.

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Chekhlov, A.R., Danchev, P.V. & Keef, P.W. Universally fully and Krylov transitive torsion-free abelian groups. Monatsh Math 198, 517–534 (2022). https://doi.org/10.1007/s00605-021-01632-7

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  • DOI: https://doi.org/10.1007/s00605-021-01632-7

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