Skip to main content
SpringerLink
Log in

Butler groups of arbitrary cardinality

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We show that in the constructible universe, the two usual definitions of Butler groups are equivalent for groups of arbitrarily large power. We also prove that Bext2(G, T) vanishes for every torsion-free groupG and torsion groupT. Furthermore, balanced subgroups of completely decomposable groups are Butler groups. These results have been known, under CH, only for groups of cardinalities ≤ ℵω.

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. U. Albrech and P. Hill,Butler groups of infinite rank and Axiom 3, Czech. Math. J.37 (1987), 293–309.

    Google Scholar 

  2. D. M. Arnold,Notes on Butler groups and balanced extensions, Boll. Unione Mat. Ital. A5 (1986), 175–184.

    MATH  Google Scholar 

  3. L. Bican,Splitting in abelian groups, Czech. Math. J.28 (1978), 356–364.

    MathSciNet  Google Scholar 

  4. L. Bican and L. Fuchs,On abelian groups by which balanced extensions of a rational group split, J. Pure Appl. Algebra78 (1992), 221–238.

    Article  MATH  MathSciNet  Google Scholar 

  5. L. Bican and L. Salce,Butler groups of infinite rank, inAbelian Group Theory, Lecture Notes in Math.1006 (Springer, 1983), 171–189.

  6. M. C. R. Butler,A class of torsion-free abelian groups of finite rank, Proc. London Math. Soc.15 (1965), 680–698.

    Article  MATH  MathSciNet  Google Scholar 

  7. M. Dugas and K. M. Rangaswamy,Infinite rank Butler groups, Trans. Amer. Math. Soc.305 (1988), 129–142.

    Article  MATH  MathSciNet  Google Scholar 

  8. M. Dugas, P. Hill and K. M. Rangaswamy,Infinite rank Butler groups, II, Trans. Amer. Math. Soc.320 (1990), 643–664.

    Article  MATH  MathSciNet  Google Scholar 

  9. M. Dugas and B. Thomé,The functor Bext under the negation of CH, Forum Math.3 (1991), 23–33.

    Article  MATH  MathSciNet  Google Scholar 

  10. P. C. Eklof and L. Fuchs,Baer modules over valuation domains, Annali Mat. Pura Appl.150 (1988), 363–374.

    Article  MATH  MathSciNet  Google Scholar 

  11. L. Fuchs,Infinite Abelian Groups, Vol. 2, Academic Press, New York, 1973.

    MATH  Google Scholar 

  12. L. Fuchs and C. Metelli,Countable Butler groups, Contemporary Math.130 (1992), 133–143.

    MathSciNet  Google Scholar 

  13. P. Hill,The third axiom of countability for abelian groups, Proc. Amer. Math. Soc.82 (1981), 347–350.

    Article  MATH  MathSciNet  Google Scholar 

  14. W. Hodges,In singular cardinality, locally free algebras are free, Algebra Universalis12 (1981), 205–220.

    Article  MATH  MathSciNet  Google Scholar 

  15. T. Jech,Set Theory, Academic Press, New York, 1973.

    MATH  Google Scholar 

  16. R. B. Jensen,The fine structure of the constructible universe, Annals of Math. Logic4 (1972), 229–308.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Tulane University, 70118, New Orleans, LA, USA

    Laszlo Fuchs

  2. Department of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel

    Menachem Magidor

  3. Department of Mathematics, Massachusetts Institute of Technology, 02139, Cambridge, MA, USA

    Menachem Magidor

Authors
  1. Laszlo Fuchs
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Menachem Magidor
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Partial support by NSF is gratefully acknowledged.

Partially supported by U.S.-Israel Binational Science Foundation.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fuchs, L., Magidor, M. Butler groups of arbitrary cardinality. Israel J. Math. 84, 239–263 (1993). https://doi.org/10.1007/BF02761702

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02761702

Keywords

Search

Navigation