On Abelian Groups Having All Proper Characteristic Subgroups Isomorphic

TL;DR

This paper investigates Abelian groups where all proper characteristic subgroups are isomorphic, exploring their properties and extending previous results on groups with fully invariant subgroups, including cases where a proper characteristic subgroup is isomorphic to the entire group.

Contribution

It provides a detailed analysis of Abelian groups with all proper characteristic subgroups isomorphic, extending earlier work and examining groups with special isomorphic subgroup properties.

Findings

Characterization of Abelian groups with all proper characteristic subgroups isomorphic.

Comparison with groups having all proper fully invariant subgroups isomorphic.

Results on groups with a proper characteristic subgroup isomorphic to the whole group.

Abstract

We consider two variants of those Abelian groups with all proper characteristic 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 subgroups isomorphic, which are studied in details by the current authors in Commun. Algebra (2015). In addition, we also examine those Abelian groups having at least one proper characteristic subgroup isomorphic to the whole group. The established by us results somewhat extend those obtained by Grinshpon-Nikolskaya in Tomsk State Univ. J. Math. & Mech. (2011, 2012) and in Commun. Algebra (2011), respectively.