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On universal locally finite groups

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Abstract

We deal with the question of existence of a universal object in the category of universal locally finite groups; the answer is negative for many uncountable cardinalities; for example, for 2ℵ 0, and assuming G.C.H. for every cardinal whose confinality is >ℵ0. However, if λ>κ when κ is strongly compact and of λ=ℵ0, then there exists a universal locally finite group of cardinality λ. The idea is to use the failure of the amalgamation property in a strong sense. We shall also prove the failure of the amalgamation property for universal locally finite groups by transferring the kind of failure of the amalgamation property from LF into ULF.

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References

  1. J. Baumgartner,Almost disjoint sets, Ann. Math. Log.10 (1975), 401–439.

    Google Scholar 

  2. D. Giorgetta and S. Shelah,Existentially closed structures in the power of the continuum, Ann. Math. Log., submitted.

  3. P. Hall,Some constructions for locally finite groups, J. London Math. Soc.34 (1959), 305–319.

    Article  MATH  MathSciNet  Google Scholar 

  4. O. Kegel and B. Wehrfritz,Locally Finite Groups, North-Holland Publ. Co., 1973.

  5. A. Macintyre and S. Shelah,Uncountable universal locally finite groups, J. Algebra43 (1976), 168–175.

    Article  MATH  MathSciNet  Google Scholar 

  6. B. H. Neumann,On Amalgam of periodic groups, Proc. R. Soc. London, Ser. A255 (1960), 477–489.

    Article  MATH  Google Scholar 

  7. S. Shelah,Categoricity in ℵ 1 of sentences of Lω1, ω (Q). Isr. J. Math.20 (1975), 127–148.

    Article  MATH  Google Scholar 

  8. S. Shelah,Classification Theory, North-Holland Publ. Co., 1978.

  9. S. Shelah,Classification theory for non-elementary classes I, J. Symb. Logic., submitted.

  10. S. Shelah,Classification theory for non-elementary classes II, J. Symb. Log., submitted.

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Authors and Affiliations

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

    Rami Grossberg & Saharon Shelah

  2. Institute for Advanced Studies, The Hebrew University of Jerusalem, Jerusalem, Israel

    Rami Grossberg & Saharon Shelah

Authors
  1. Rami Grossberg
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  2. Saharon Shelah
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Additional information

We would like to thank Simon Thomas for reading carefully a preliminary version of this paper, proving Lemma 20 and making valuable remarks. Also we thank the United States—Israel Binational Science Foundation for partially supporting this work.

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Grossberg, R., Shelah, S. On universal locally finite groups. Israel J. Math. 44, 289–302 (1983). https://doi.org/10.1007/BF02761988

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  • DOI: https://doi.org/10.1007/BF02761988

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