On Abelian Groups Having Isomorphic Proper Strongly Invariant Subgroups

TL;DR

This paper investigates Abelian groups where all proper strongly invariant subgroups are isomorphic, comparing their properties to groups with other types of invariant subgroups, and explores groups with a proper strongly invariant subgroup isomorphic to the whole group.

Contribution

It provides a detailed analysis of Abelian groups with all proper strongly invariant subgroups isomorphic, expanding understanding in this specific area of group theory.

Findings

Characterization of Abelian groups with all proper strongly invariant subgroups isomorphic

Comparison with groups having fully invariant or characteristic subgroups

Identification of groups with a proper strongly invariant subgroup isomorphic to the whole group

Abstract

We consider two variants of those Abelian groups with all proper strongly invariant subgroups isomorphic and give an in-depth study of their basic and specific properties in either parallel or contrast to the Abelian groups with all proper fully invariant (respectively, characteristic) subgroups isomorphic, which are studied in details by the current authors in Commun. Algebra (2015) and in J. Commut. Algebra (2023). In addition, we also explore those Abelian groups having at least one proper strongly invariant subgroup isomorphic to the whole group.