Supper Bassian and Nearly Generalized Bassian Abelian Groups

TL;DR

This paper investigates the properties of Bassian and generalized Bassian abelian groups, providing characterizations of nearly generalized Bassian groups and super Bassian groups, and clarifying hereditary properties.

Contribution

It offers new characterizations of nearly generalized Bassian and super Bassian groups, extending previous results and clarifying hereditary properties of Bassian groups.

Findings

Nearly generalized Bassian groups are either quasi-cyclic or of finite torsion-free rank.

Complete description and characterization of super Bassian groups.

Hereditary Bassian property coincides with the standard Bassian property.

Abstract

In connection to two recent publications of ours in Arch. Math. Basel (2021) and Acta Math. Hung. (2022), respectively, and in regard to the results obtained in Arch. Math. Basel (2012), we have the motivation to study the near property of both Bassian and generalized Bassian groups. Concretely, we prove that if an arbitrary reduced group has the property that all of its proper subgroups are generalized Bassian, then it is also generalized Bassian itself, that is, if G is an arbitrary nearly generalized Bassian group, then either G is a quasi-cyclic group or G is generalized Bassian of finite torsion-free rank. Moreover, we show that there is a complete description of the so-called super Bassian groups, that are those groups whose epimorphic images are all Basian, and give their full characterization. Also, we establish that the hereditary property of Bassian groups gives nothing new as it coincides with the ordinary Bassian property.