Poor and pi-poor Abelian groups

Abstract

In this paper poor abelian groups are characterized. It is proved that an abelian group is poor if and only if its torsion part contains a direct summand isomorphic to http://www.w3.org/1999/xlink where P is the set of prime integers. We also prove that pi-poor abelian groups exist. Namely it is proved that the direct sum of U-(N) where U ranges over all nonisomorphic uniform abelian groups is pi-poor. Moreover for a pi-poor abelian group M it is shown that M can not be torsion and each p-primary component of M is unbounded. Finally we show that there are pi-poor groups which are not poor and vise versa.