  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]).
