Skip to main content
SpringerLink
Log in

Abelian Groups Isomorphic to a Proper Fully Invariant Subgroup

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

The present paper is a survey of the authors’ results related to studying groups containing a proper fully invariant subgroup isomorphic to the group.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. A. Beaumont, “Groups with isomorphic proper subgroups,” Bull. Amer. Math. Soc., 51, 381–387 (1945).

    Article  MathSciNet  Google Scholar 

  2. R. A. Beaumont, “Partly transitive modules and modules with proper isomorphic submodules,” Trans. Amer. Math. Soc., 91, 209–219 (1959).

    Article  MathSciNet  Google Scholar 

  3. R. A. Beaumont and R. S. Pierce, “Isomorphic direct summands of Abelian groups,” Math. Ann., 153, 21–37 (1964).

    Article  MathSciNet  Google Scholar 

  4. K. M. Benabdallah, B. J. Eisenstadt, J. M. Irwin, and E. W. Poluianov, “The structure of large subgroups of primary Abelian groups,” Acta Math. Acad. Sci. Hungar., 21, No. 3-4, 421–435 (1970).

    Article  MathSciNet  Google Scholar 

  5. P. Crawley, “An infinite primary Abelian group without proper isomorphic subgroups,” Bull. Amer. Math. Soc., 68, 463–467 (1962).

    Article  MathSciNet  Google Scholar 

  6. L. Fuchs, Infinite Abelian Groups, Vol. I, Academic Press, London (1970).

  7. L. Fuchs, Infinite Abelian Groups, Vol. II, Academic Press, London (1973).

  8. B. Goldsmith, S. Óhógáin, and S. Wallutis, “Quasi-minimal groups,” Proc. Amer. Math. Soc., 132, No. 8, 2185–2195 (2004).

    Article  MathSciNet  Google Scholar 

  9. S. Ya. Grinshpon, “On some classes of primary Abelian groups almost isomorphic by fully invariant subgroups,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 2, 23–30 (1976).

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

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

    MathSciNet  MATH  Google Scholar 

  12. S. Ya. Grinshpon and P. A. Krylov, “Fully invariant subgroups, full transitivity and homomorphism groups of Abelian groups,” J. Math. Sci., 128, No. 3, 2894–2997 (2005).

    Article  MathSciNet  Google Scholar 

  13. S. Ya. Grinshpon and M. M. Nikolskaya (Savinkova), “IF-groups,” Tomsk State Univ. J. Math. Mech., No. 1 (9), 5–14 (2010).

  14. S. Ya. Grinshpon and M. M. Nikolskaya, “Abelian p-groups non-isomorphic to proper fully invariant subgroups,” in: Int. Conf. “Algebra, Logic and Applications” (Krasnoyarsk, July, 19–25, 2010 ) [in Russian], pp. 27–28.

  15. S. Ya. Grinshpon and M. M. Nikolskaya, “Proper fully invariant subgroups isomorphic to the group,” in: Abelian Groups: All-Russia Symposium on the Occasion of the 95th Anniversary of L. Ya. Kulikov (Biysk, August, 19–25, 2010 ) [in Russian], pp. 29–31.

  16. S. Ya. Grinshpon and M. M. Nikolskaya (Savinkova), “Fully invariant subgroups of Abelian p-groups with finite Ulm–Kaplansky invariants,” Commun. Algebra, 39, Issue 11, 4273–4282 (2011).

  17. S. Ya. Grinshpon and M. M. Nikolskaya, “Primary IF-groups,” Tomsk State Univ. J. Math. Mech., No. 3 (15), 25–31 (2011).

  18. S. Ya. Grinshpon and M. M. Nikolskaya, “Proper fully invariant subgroups of torsion free groups isomorphic to the group,” Tomsk State Univ. J. Math. Mech., No. 1 (17), 11–15 (2012).

  19. P. Hill and Ch. Megibben, “On primary groups with countable basic subgroups,” Trans. Amer. Math. Soc., 124, No. 1, 49–59 (1966).

    Article  MathSciNet  Google Scholar 

  20. I. Kaplansky, Infinite Abelian Groups, Univ. Michigan Press, Ann Arbor (1968).

  21. L. Ya. Kulikov, “On the theory of Abelian groups of arbitrary power,” Mat. Sb., No. 9, 165–182 (1941).

  22. L. Ya. Kulikov, “On the theory of Abelian groups of arbitrary power,” Mat. Sb., No. 16, 129–162 (1945).

  23. G. S. Monk, “Abelian p-groups without proper isomorphic pure dense subgroups,” Illinois J. Math., 14, No. 1, 164–177 (1970).

    Article  MathSciNet  Google Scholar 

  24. R. S. Pierce, “Homomorphisms of primary Abelian groups,” in: Topics in Abelian Groups, Chicago (1963), pp. 215–310.

  25. M. M. Savinkova, “Fully invariant subgroups of Abelian p-groups, isomorphic to the group,” in: Int. Algebraic Conf. on the Occasion of the 100th Anniversary of Professor A. G. Kurosh (1908–1971) (Moscow, May 28–June 3, 2008 ), pp. 200–201.

  26. M. M. Savinkova, “On separable p-groups containing fully invariant subgroups isomorphic to the group,” in: All-Russia Conf. on Mathematics and Mechanics (Tomsk, September 22–25, 2008 ) [in Russian], p. 62.

  27. M. M. Savinkova, “U-sequences and primary groups having proper fully invariant subgroups isomorphic them,” Tomsk State Univ. J. Math. Mech., No. 2 (3), 56–60 (2008).

  28. A. I. Sherstneva (Botygina), “U-sequences and almost isomorphism of Abelian p-groups by fully invariant subgroups,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 5, 72–80 (2001).

Download references

Author information

Authors and Affiliations

  1. Tomsk State University, Tomsk, Russia

    S. Ya. Grinshpon

  2. Tomsk State University of Architecture and Building, Tomsk, Russia

    M. M. Nikolskaya

Authors
  1. S. Ya. Grinshpon
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. M. Nikolskaya
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Ya. Grinshpon.

Additional information

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 22, No. 5, pp. 29–53, 2019.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Grinshpon, S.Y., Nikolskaya, M.M. Abelian Groups Isomorphic to a Proper Fully Invariant Subgroup. J Math Sci 259, 403–419 (2021). https://doi.org/10.1007/s10958-021-05629-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-021-05629-3

Search

Navigation