Poor and pi-poor abelian groups

TL;DR

This paper characterizes poor abelian groups, introduces the concept of pi-poor abelian groups, and explores their properties, including existence, structure, and differences from poor groups.

Contribution

It provides the first characterization of poor abelian groups and proves the existence of pi-poor abelian groups, highlighting their structural properties and distinctions from poor groups.

Findings

An abelian group is poor iff its torsion part contains a direct summand isomorphic to a direct sum of p-adic integers.

Pi-poor abelian groups exist, exemplified by a specific direct sum of uniform groups.

Pi-poor groups cannot be torsion, and their p-primary components are unbounded.

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 $\oplus_{p \in P} \Z_p$, 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.