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 ${\oplus }_{p\in P}{\mathbb{Z}}_p$, 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 $U^{(\mathbb{N})}$, 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.

The necessity part of Theorem 3.1. is proved without using basic subgroups