On the generalized Bassian property for Abelian groups

Generalizations of fully transitive and valuated Abelian p-groups

“…We now arrive at our first chief result which expand the corresponding closeness of the direct summand property for both Bassian and generalized Bassian groups (compare with [1] and [2]).…”

Abelian p-groups with minimal characteristic inertia

“…In Section 3 we prove that, as with Bassian groups, any generalized Bassian group necessarily has finite torsion-free rank (Theorem 3.5). This gives an affirmative answer to a question from [3] (Section 4). Second, we show that any group G with finite torsion-free rank and bounded p-torsion must necessarily be isomorphic to a direct sum H ⊕ S, where H is a Bassian group and S is torsion (Theorem 3.1).…”

“…Second, we show that any group G with finite torsion-free rank and bounded p-torsion must necessarily be isomorphic to a direct sum H ⊕ S, where H is a Bassian group and S is torsion (Theorem 3.1). This gives much more direct connection between the classes of Bassian and generalized Bassian than was apparent in [3].…”

“…Definition 1.1. ( [2], [3]) A group G is said to be Bassian if it cannot be embedded in a proper homomorphic image of itself or, equivalently, if the existence of an injection G → G/N, for some subgroup N of G, forces that N = {0}.…”

“…Restated slightly, the Main Theorem of [2] is that G is Bassian if, and only if, it has finite torsion-free rank and, for all primes p, T p is also finite. The generalized Bassian groups were not fully characterized in [3], but it was shown that if G is generalized Bassian, then pT p must be finite for all primes p ([3], Lemma 3.1). The main goal of this paper is to describe the class of generalized Bassian groups in greater detail.…”