Skip to main content

Determinateness of Torsion-Free Abelian Groups by Their Holomorphs and Almost Holomorphic Isomorphism

Abstract

In this paper, we study holomorphically isomorphic Abelian groups, i.e., Abelian groups with isomorphic holomorphs. We also study a generalization of the concept of holomorphic isomorphism, namely, almost holomorphic isomorphism, which is deeply connected with normal Abelian subgroups of holomorphs of Abelian groups. Torsion-free Abelian groups that are determined by their holomorphs are highlighted from different classes. In particular, it has been found that any homogeneous separable group can be determined by its holomorph in the class of all Abelian groups.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    I. Kh. Bekker, “On holomorphs of Abelian groups,” Sib. Mat. Zh., 5, No. 6, 1228–1238 (1964).

    MATH  MathSciNet  Google Scholar 

  2. 2.

    I. Kh. Bekker, “On holomorphs of non-reduced Abelian groups,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 8, 3–8 (1968).

  3. 3.

    I. Kh. Bekker, “On holomorphs of torsion-free Abelian groups,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 3, 3–13 (1974).

  4. 4.

    I. Kh. Bekker, “Abelian groups with isomorphic holomorphs,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 3, 97–99 (1975).

  5. 5.

    I. Kh. Bekker, “Abelian holomorphic groups,” in: Int. Conf. “All-Siberian Readings on Mathematics and Mechanics.” Selected Reports. V. 1. Mathematics [in Russian] (1997), pp. 43–47.

  6. 6.

    I. Kh. Bekker and S. Ya. Grinshpon, “Almost holomorphically isomorphic primary Abelian groups,” in: Groups and Modules. Interacademic Subject Collection of Scientific Works (1976), pp. 90–103.

  7. 7.

    L. Fuchs, Infinite Abelian Groups, V. 2 [Russian translation], Mir, Moscow (1977).

  8. 8.

    I. E. Grinshpon, “Normal subgroups of holomorphs of Abelian groups and almost holomorphic isomorphism,” Fundam. Prikl. Mat., 13, No. 3, 9–16 (2007).

    Google Scholar 

  9. 9.

    S. Ya. Grinshpon, “Almost holomorphically isomorphic Abelian groups,” Tr. Tomsk. Univ., 220, Problems of Mathematics, No. 3, 78–84 (1975).

  10. 10.

    S. Ya. Grinshpon, “On the structure of fully invariant subgroups of torsion-free Abelian groups,” in: Abelian Groups and Modules, Tomsk (1982), pp. 56–92.

  11. 11.

    S. Ya. Grinshpon, “Fully invariant subgroups of Abelian groups and full transitivity,” Fundam. Prikl. Mat., 8, No. 2, 407–473 (2002).

    MATH  MathSciNet  Google Scholar 

  12. 12.

    G. A. Miller, “On the multiple holomorph of a group,” Math. Ann., 66, 133–142 (1908).

    Article  MathSciNet  Google Scholar 

  13. 13.

    W. H. Mills, “Multiple holomorphs of finitely generated Abelian groups,” Trans. Am. Math. Soc., 71, No. 3, 379–392 (1950).

    Article  Google Scholar 

  14. 14.

    W. H. Mills, “On the non-isomorphism of certain holomorphs,” Trans. Am. Math. Soc., 74, No. 3, 428–443 (1953).

    Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to S. Ya. Grinshpon.

Additional information

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 17, No. 8, pp. 35–46, 2011/12.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Grinshpon, S.Y., Grinshpon, I.E. Determinateness of Torsion-Free Abelian Groups by Their Holomorphs and Almost Holomorphic Isomorphism. J Math Sci 197, 605–613 (2014). https://doi.org/10.1007/s10958-014-1742-1

Download citation

Keywords

  • Abelian Group
  • Normal Subgroup
  • Nonzero Element
  • Homogeneous Group
  • Direct Summand