Israel Journal of Mathematics

, Volume 60, Issue 2, pp 146–166 | Cite as

On reconstructing separable reducedp-groups with a given socle

  • Saharon Shelah
Article
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Abstract

Let\(\bar B^* \) be a separable reduced (abelian)p-group which is torsion complete. We ask whether for\(G \subseteq \bar B^* \) there is\(H \subseteq _{pr} \bar B^* ,H[p] = G[p]\),H[p]=G[p],H not isomorphic toG. IfG is the sum of cyclic groups or is torsion complete, the answer is easily no. For otherG, we prove that the answer is yes assuming G.C.H. Even without G.C.H. the answer is yes if the density character ofG is equal to Min n<ω|p nG|, i.e.,
$$\mathop {Min}\limits_{n< \omega } |p^n G| = \mathop {Min}\limits_m \mathop \Sigma \limits_{n > m} |(p^n G)[p]/(p^{n + 1} G)[p]|$$
Of course, instead of two non-isomorphic we can get many, but we do not deal much with this.

Keywords

Free Algebra Regular Cardinal Density Character Pure Subgroup Singular Cardinal 
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. [BD]
    S. Ben-David,On Shelah’s compactness of cardinals, Isr. J. Math.31 (1978), 34–56; 394.MATHCrossRefMathSciNetGoogle Scholar
  2. [F]
    L. Fuchs,Abelian Groups, Vols. 1, 2, Academic Press, New York, 1970, 1973.Google Scholar
  3. [G-S]
    R. Grossberg and S. Shelah,On universal locally finite groups, Isr. J. Math.44 (1983), 289–302.MATHCrossRefMathSciNetGoogle Scholar
  4. [H]
    W. Hodges,In singular cardinality, locally free algebras are free, Algebra Univ.12 (1981), 205–220.MATHCrossRefMathSciNetGoogle Scholar
  5. [Mk-Sh]
    A. Mekler and S. Shelah,Uniformization principles, J. Symb. Logic, to appear.Google Scholar
  6. [Sh 1]
    S. Shelah,A compactness theorem in singular cardinals, free algebras, Whitehead problems and transversal, Isr. J. Math.21 (1975), 319–344.MATHCrossRefGoogle Scholar
  7. [Sh 2]
    S. Shelah,Incompactness in regular cardinals, Notre Dame J. Formal Logic26 (1985), 195–228.MATHCrossRefMathSciNetGoogle Scholar
  8. [Sh 3]
    S. Shelah,Classification Theory, North-Holland, Amsterdam, 1978.MATHGoogle Scholar
  9. [Sh 4]
    S. Shelah,Constructions of many complicated uncountable structures and Boolean algebras, Isr. J. Math.45 (1983), 100–146.MATHCrossRefGoogle Scholar
  10. [Sh 5]
    S. Shelah,Proper Forcing, Lecture Notes No. 940, Springer-Verlag, Berlin, 1982.MATHGoogle Scholar

Copyright information

© The Weizmann Science Press of Israel 1987

Authors and Affiliations

  • Saharon Shelah
    • 1
    • 2
  1. 1.The Hebrew University of JerusalemJerusalemIsrael
  2. 2.Simon Fraser UniversityBurnabyCanada

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