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Hereditarily separable groups and monochromatic uniformization


We give a combinatorial equivalent to the existence of a non-free hereditarily separable group of cardinality ℵ1. This can be used, together with a known combinatorial equivalent of the existence of a non-free Whitehead group, to prove that it is consistent that every Whitehead group is free but not every hereditarily separable group is free. We also show that the fact that ℤ is a p.i.d. with infinitely many primes is essential for this result.

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  1. [1]

    U. Avraham and S. Shelah,Forcing with stable posets, Journal of Symbolic Logic47 (1982), 37–42.

  2. [2]

    K. Devlin and S. Shelah, A weak version of\(2^{\aleph _0 }< 2^{\aleph _0 } \) which follows from 2N 0<2N 1, Israel Journal of Mathematics29 (1978), 239–247.

  3. [3]

    P. C. Eklof and A. H. Mekler,Free Modules, North-Holland, Amsterdam, 1990.

  4. [4]

    P. C. Eklof, A. H. Mekler and S. Shelah,Uniformization and the diversity of Whitehead groups, Israel Journal of Mathematics80 (1992), 301–321.

  5. [5]

    H. P. Goeters and W. J. Wickless,Hyper-τ groups, Communications in Algebra17 (1989), 1275–1290.

  6. [6]

    F. Richman,A class of rank-2 torsion free groups, inStudies on Abelian Groups (B. Charles, ed.), Springer-Verlag, Berlin, 1968, pp. 327–334.

  7. [7]

    S. Shelah,Infinite abelian groups, Whitehead problem and some constructions, Israel Journal of Mathematics18 (1974), 243–256.

  8. [8]

    S. Shelah,Whitehead groups may not be free even assuming CH, I, Israel Journal of Mathematics28 (1977), 193–203.

  9. [9]

    S. Shelah,On uncountable abelian groups, Israel Journal of Mathematics32 (1979), 311–330.

  10. [10]

    S. Shelah,Whitehead groups may not be free even assuming CH, II, Israel Journal of Mathematics35 (1980), 257–285.

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Author information

Correspondence to P. C. Eklof.

Additional information

The authors thank Rutgers University for its support.

Research partially supported by NSERC grant #9848. Prof. Mekler died on June 10, 1992.

Research partially supported by the BSF. Publication #442.

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Eklof, P.C., Mekler, A.H. & Shelah, S. Hereditarily separable groups and monochromatic uniformization. Israel J. Math. 88, 213–235 (1994).

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