### 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]).

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