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Israel Journal of Mathematics

, Volume 18, Issue 3, pp 243–256 | Cite as

Infinite abelian groups, whitehead problem and some constructions

  • Saharon Shelah
Article

Abstract

We solve here some problems from Fuchs’ book. We show that the answer to Whitehead’s problem (for groups of power ℵ1) is independent from the usual axioms of set theory. We prove the existence of large rigid systems for groups of power λ, with no restriction on λ. We also prove that there are many non-isomorphic reduced separablep-groups.

Keywords

Abelian Group Regular Cardinal Measurable Cardinal Stationary Subset Pure Subgroup 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    P. J. Cohen,Set theory and the continuum hypothesis, W. A. Benjamin, Inc., 1966.Google Scholar
  2. 2.
    P. Eklof,Infinitary equivalence of abelian groups, Fund. Math., to appear.Google Scholar
  3. 3.
    P. Eklof,On the existence of k-free abelian groups.Google Scholar
  4. 4.
    P. Eklof,Theorems of ZFCon abelian groups infinitarily equivalent to free groups, Notices Amer. Math. Soc.20 (1973), A-503.Google Scholar
  5. 5.
    P. Erdös and R. Rado,Intersection theorems for systems of sets, J London Math. Soc.44 (1969), 467–479.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    G. Fodor,Eine Bemerdeeng zur Theorie der regressiven Funktionen, Acta. Sci. Math.17 (1956), 139–142.zbMATHMathSciNetGoogle Scholar
  7. 7.
    L. Fuchs,Infinite abelian groups, Vol. I, Academic Press, N. Y. & London, 1970.zbMATHGoogle Scholar
  8. 8.
    L. Fuchs,Infinite abelian groups, Vol. II, Academic Press, N. Y. & London, 1973.zbMATHGoogle Scholar
  9. 9.
    L. Fuchs,Abelian Groups, Publishing house of the Hungarian Academy of Sciences, Budapest, 1958.zbMATHGoogle Scholar
  10. 10.
    L. Fuchs,Indecomposable abelian groups of measurable cardinals dedicated to R. Baer, to appear.Google Scholar
  11. 11.
    K. Godel,The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory, Princeton University Press, Princeton, N. J., 1940.Google Scholar
  12. 12.
    J. Gregory,Abelian groups infinitarily equivalent to free ones, Notices Amer. Math. Soc.20, 1973, A-500.Google Scholar
  13. 13.
    E. Hewitt and K. A. Ross,Abstract Harmonic Analysis.Google Scholar
  14. 14.
    R. B. Jensen,The fine structure of the constructible hierarchy, Ann. Math. Logic4 (1972), 229–308.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    R. B. Jensen, K. Kunen,Some combinatorial properties of V and L, Notes.Google Scholar
  16. 16.
    A. Mekler, Ph.D. thesis, Stanford Univ., in preparation.Google Scholar
  17. 17.
    D. M. Martin, R. M. Solovay,Internal Cohen extension, Ann. Math. Logic2 (1970), 143–178.zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    R.S. Pierce,Homomorphisms of primary abelian groups, topics in abelian groups, Chicago, Illinois, 1963, pp. 215–310.Google Scholar
  19. 19.
    J. Rotman,On a problem of Baer and a problem of Whitehead in abelian groups, Acta. Math. Acad. Sci. Hungar.12 (1961), 245–254.zbMATHCrossRefGoogle Scholar
  20. 20.
    S. Shelah,Categoricity of uncountable theories, Proc. of Tarski Symp., Berkeley 1971 to appear.Google Scholar
  21. 21.
    R. M. Solovay,Real-valued measurable cardinals, Proceedings of Symposia in Pure Mathematics, XIII, Part I, Amer. Math. Soc., Providence, R. I., 1971.Google Scholar
  22. 22.
    K. Stein,Analytische Funktionen mehrerer komplexer Veranderlichen zer vorgegebenen Periodizitatsmoduln und das ziveite Cousinshe Problem, Math. Ann.123 (1951), 201–222.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Hebrew University 1974

Authors and Affiliations

  • Saharon Shelah
    • 1
  1. 1.Institute of MathematicsThe Hebrew University of JerusalemJerusalemIsrael

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