GCH implies the existence of many rigid almost free abelian groups

Details GCH implies the existence of many rigid almost free abelian groups GCH implies the existence of many rigid almost free abelian groups

2000-11-22

arxiv.org/abs/math/0011185v1

Proceedings of the Conference on Abelian Groups in Colorado Springs (1995), volume 182 of Lecture Notes in Pure and Applied Ma

Mathematics

Group Theory

Scientific paper

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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