G.C.H. implies existence of many rigid almost free abelian groups

by Goebel and Shelah. [GbSh:579]
Proc Conference on Abelian Groups in Colorado Springs, 1995, 1996
We begin with the existence of groups with trivial duals for cardinals aleph_n (n in omega). Then we derive results about strongly aleph_n-free abelian groups of cardinality aleph_n (n in omega) with prescribed free, countable endomorphism ring. Finally we use combinatorial results of [Sh:108], [Sh:141] to give similar answers for cardinals > aleph_omega . As in Magidor and Shelah [MgSh:204], a paper concerned with the existence of kappa-free, non-free abelian groups of cardinality kappa, the induction argument breaks down at aleph_omega . Recall that aleph_omega is the first singular cardinal and such groups of cardinality aleph_omega do not exist by the well-known Singular Compactness Theorem (see [Sh:52]).


Back to the list of publications