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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 $G/H\cong G/K$, 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...