Hereditarily and Super Generalized co-Bassian Abelian Groups

TL;DR

The paper addresses complete characterization of hereditary and super properties for co-Bassian and generalized co-Bassian Abelian groups. It builds on the established characterization (K), linking $T_p$-ranks and the structure of $G/T$ to divisibility and finite rank conditions. For hereditary co-Bassian groups, the authors prove that $G$ is hereditarily co-Bassian iff $G$ has no subgroup isomorphic to $\mathbb{Z}$ or to $\mathbb{Z}(p)^{(\omega)}$ for any prime $p$, equivalently $G$ is a torsion group with each $T_p$ finitely co-generated. For hereditary generalized co-Bassian groups, the corresponding criterion replaces $\mathbb{Z}(p)^{(\omega)}$ with $\mathbb{Z}(p^2)^{(\omega)}$ and requires that $pT_p$ be finitely co-generated for all primes $p$. The paper then provides a criterion for when a group is super generalized co-Bassian: either $G$ is divisible, or $G/T$ is torsion-free divisible of finite rank and, for each prime $p$, $T_p$ is divisible or $pT_p$ has finite $p$-rank. Together, these results clarify the subgroups and quotient-structures that govern these classes and connect to the known (K) characterizations.

Abstract

We completely characterize by finding necessary and sufficient conditions those co-Bassian and generalized co-Bassian Abelian groups having, respectively, the hereditary or the super property, thus giving a new insight in the full discovery of the structure of these two classes of groups as recently defined in Arch. Math. Basel (2024) by the third author.

Theorems & Definitions (12)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Lemma 2.3
• proof
• Theorem 2.4
• proof