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Hereditarily separable groups and monochromatic uniformization

Abstract

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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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). https://doi.org/10.1007/BF02937512

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Keywords

  • Direct Summand
  • Israel Journal
  • Separable Group
  • Torsion Free Group
  • Regular Cardinal