Semi-Generalized Bassian Groups

TL;DR

This paper introduces semi-generalized Bassian groups, providing their complete characterization in various cases and exploring related subgroup properties, advancing the understanding of this new class of groups.

Contribution

It defines semi-generalized Bassian groups and offers a comprehensive classification in p-torsion, torsion-free, and splitting mixed groups, expanding group theory knowledge.

Findings

Complete characterization of semi-generalized Bassian groups in specific cases

Classification of groups with every subgroup essential in a direct summand

Extension of Bassian group concepts to broader classes

Abstract

As a common non-trivial generalization of the concept of a proper generalized Bassian group, we introduce the notion of a semi-generalized Bassian group and initiate its comprehensive investigation. Precisely, we give a satisfactory characterization of these groups by showing in the cases of p-torsion groups, torsion-free groups and splitting mixed groups their complete description. Moreover, we classify the groups with the clearly related property that every subgroup is essential in a direct summand of the whole group.