Some generalizations of torsion-free Crawley groups

Abstract

In this paper we investigate two new classes of torsion-free Abelian groups which arise in a natural way from the notion of a torsion-free Crawley group. A group G is said to be an Erdős group if for any pair of isomorphic pure subgroups H,K with G/H ⋟ G/K, there is an automorphism of G mapping H onto K; it is said to be a weak Crawley group if for any pair H,K of isomorphic dense maximal pure subgroups, there is an automorphism mapping H onto K. We show that these classes are extensive and pay attention to the relationship of the Baer-Specker group to these classes. In particular, we show that the class of Crawley groups is strictly contained in the class of weak Crawley groups and that the class of Erdős groups is strictly contained in the class of weak Crawley groups.