On Abelian Groups Having All Proper Fully Invariant Subgroups Isomorphic

Abstract

We introduce two classes of abelian groups which have either only trivial fully invariant subgroups or all their nontrivial (respectively nonzero) fully invariant subgroups are isomorphic, called IFI-groups and strongly IFI-groups, such that every strongly IFIgroup is an IFI-group, respectively. Moreover, these classes coincide when the groups are torsion-free, but are different when the groups are torsion as well as, surprisingly, mixed groups cannot be IFI-groups. We also study their important properties as our results somewhat contrast with those from [13] and [14].

On Abelian Groups Having All Proper Characteristic Subgroups Isomorphic

“…To prove necessity, let H be characteristic in G with H = G. Since G is a weakly IC-group and H ∼ = G, we deduce that H as well as G is unbounded. But then H ∩ B is characteristic in B and H ∩ B is basic in H in accordance with Lemma 2.37 (2). Note that H ∩ B = B, because if we assume that B ≤ H, then in view of the equality B + H = G it would follow that H = G -a contradiction.…”

“…And so, we firstly will comment the Abelian groups with an isomorphic proper characteristic subgroup, which groups were termed here as weakly ICgroups. Some of the analogies and differences with the classes of IC-groups and strongly IC-groups, both introduced in [2], are as follows: (1) These three classes are definitely not closed under taking direct summands; (2) They have totally different structure as shown above.…”

“…• If G is a torsion-free group, then G is an IC-group if, and only if, G is a strongly IC-group (compare with the next subsection). The proof is as Proposition 2.2 from [2].…”

“…For completeness of the exposition and for the reader's convenience, we first and foremost will give a brief retrospection of the most principal results achieved in [2] concerning IFI-groups and strongly IFI-groups.…”