GCH implies the existence of many rigid almost free abelian groups
Group Theory
2007-05-23 v1 Logic
Abstract
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]).
Cite
@article{arxiv.math/0011185,
title = {GCH implies the existence of many rigid almost free abelian groups},
author = {Rüdiger Göbel and Saharon Shelah},
journal= {arXiv preprint arXiv:math/0011185},
year = {2007}
}