GCH implies the existence of many rigid almost free abelian groups

TL;DR

This paper explores the existence and properties of certain large abelian groups with specific dual and endomorphism characteristics, extending results to singular cardinals like aleph_omega using combinatorial set theory.

Contribution

It establishes new results on the existence of strongly free abelian groups with prescribed endomorphism rings at various infinite cardinals, including singular ones.

Findings

Existence of groups with trivial duals at aleph_n for n in omega.

Construction of strongly aleph_n-free abelian groups with prescribed endomorphism rings.

Extension of results to cardinals greater than aleph_omega using combinatorial methods.

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]).