ON ABELIAN GROUPS HAVING ISOMORPHIC PROPER CHARACTERISTIC SUBGROUPS

Abstract

We consider two variants of Abelian groups where (all) proper characteristic subgroups are isomorphic and give an in-depth study of their basic and specific properties in either parallel or contrast to the Abelian groups where all proper fully invariant subgroups are isomorphic, which are studied in details by the 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. We prove in these directions, by the use of concrete terminology, that any basic subgroup of a $p$-primary separable weakly $IC$-group remains a weakly $IC$-group, as well as that a torsion-complete $p$-group is a weakly $IC$-group if and only if some of its basic subgroups are weakly $IC$-groups. These results extend those obtained by Grinshpon and Nikolskaya in Tomsk State Univ. J. Math. & Mech. (2011, 2012) and in Commun. Algebra (2011), respectively.