Generalized Bassian and other Mixed Abelian Groups with Bounded p-Torsion

TL;DR

This paper characterizes generalized Bassian abelian groups with bounded p-torsion, proving they must have finite torsion-free rank and are B+E-groups, thus advancing understanding of their structure.

Contribution

It provides a complete characterization of generalized Bassian groups with bounded p-torsion, answering a recent open question and establishing their structural properties.

Findings

All generalized Bassian groups have finite torsion-free rank.

Every generalized Bassian group is a B+E-group, i.e., a sum of a Bassian and an elementary group.

A subgroup of a generalized Bassian group is itself B+E.

Abstract

It is known that a mixed abelian group G with torsion T is Bassian if, and only if, it has finite torsion-free rank and has finite p-torsion (i.e., each Tp is finite). It is also known that if G is generalized Bassian, then each pTp is finite, so that G has bounded p-torsion. To further describe the generalized Bassian groups, we start by characterizing the groups in some important classes of mixed groups with bounded p-torsion (e.g., the balanced-projective groups and the Warfield groups). We then prove that all generalized Bassian groups must have finite torsion-free rank, thus answering a question recently posed in Acta Math. Hung. (2022) by Chekhlov-Danchev-Goldsmith. This implies that every generalized Bassian group must be a B+E-group; i.e., the direct sum of a Bassian group and an elementary group. The converse is shown to hold for a large class of mixed groups, including the Warfield groups. It is also proved that G is a B+E-group if, and only if, it is a subgroup of a generalized Bassian group.