Skip to main content
SpringerLink
Log in

On some generalizations of Hopfian and co-Hopfian Abelian groups

  • Published:
Acta Mathematica Hungarica Aims and scope Submit manuscript

Abstract

The notions of Hopfian and co-Hopfian groups have been of interest for some time. In this present work we consider generalizations of the classes of hereditarily Hopfian (co-Hopfian) and super Hopfian (co-Hopfian) Abelian groups by requiring only that countable subgroups or images inherit the Hopfian (co-Hopfian) property.

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. Baer, Groups without proper isomorphic quotient groups, Bull. Amer. Math. Soc., 50 (1944), 267–278.

    Article  MathSciNet  MATH  Google Scholar 

  2. G. Braun and L. Strüngmann, The independence of the notions of Hopfian and co-Hopfian Abelian p-groups, to appear.

  3. A. L. S. Corner, Extensions of torsion groups by countable torsion-free groups, unpublished manuscript [U14] listed in [6]; to appear at website http://arrow.dit.ie/cgi/siteview.cgi/corner.

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

    Article  MathSciNet  MATH  Google Scholar 

  5. L. Fuchs, Infinite Abelian Groups, Vols. I and II, Academic Press (New York, 1970 and 1973).

    Google Scholar 

  6. B. Goldsmith, Anthony Leonard Southern Corner 1934–2006, in: R. Göbel and B. Goldsmith (Eds.) Models, Modules and Abelian Groups, Walter de Gruyter (Berlin, 2008), pp. 1–7.

    Google Scholar 

  7. B. Goldsmith and K. Gong, A note on Hopfian and co-Hopfian Abelian groups, in: M. Droste, L. Fuchs, L. Strüngmann, K. Tent and M. Ziegler (Eds.) Contemporary Mathematics 576 (2012), pp. 129–136.

    Google Scholar 

  8. B. Goldsmith and K. Gong, On super and hereditarily Hopfian and co-Hopfian Abelian groups, Archiv der Math., 99 (2012), to appear.

  9. B. Goldsmith and K. Gong, On adjoint entropy of Abelian groups, Comm. Algebra, 40 (2012), 972–987.

    Article  MathSciNet  MATH  Google Scholar 

  10. J. D. Reid, A note on torsion-free Abelian groups of infinite rank, Proc. Amer. Math. Soc., 13 (1962), 222–225.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Dublin Institute of Technology, Aungier Street, Dublin, 2, Ireland

    B. Goldsmith

  2. School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin, 8, Ireland

    K. Gong

Authors
  1. B. Goldsmith
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. Gong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B. Goldsmith.

Additional information

Corresponding author.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Goldsmith, B., Gong, K. On some generalizations of Hopfian and co-Hopfian Abelian groups. Acta Math Hung 139, 393–398 (2013). https://doi.org/10.1007/s10474-012-0290-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10474-012-0290-8

Key words and phrases

Mathematics Subject Classification

Search

Navigation