Generalized Bassian and other mixed Abelian groups with bounded p-torsion

Abstract

It is known that a mixed Abelian group G with torsion subgroup T is Bassian if, and only if, it has finite torsion-free rank and has finite each p-torsion component Tp (i.e., Tp is finite for all primes p). It is also known that if G is generalized Bassian, then each pTp is finite, so that G has bounded p-torsion. To further describe the generalized Bassian groups, we start by characterizing the groups in some important classes of mixed groups (e.g., the balanced-projective groups and the Warfield groups) having bounded p-torsion. We then prove that all generalized Bassian groups must have finite torsion-free rank, thus answering a question recently posed by Chekhlov, Danchev, Goldsmith (2022) [3]. This implies that every generalized Bassian group must be a B+E-group; that is, the direct sum of a Bassian group and an elementary group. The converse is shown to hold for a large class of mixed groups, including the Warfield groups. It is also proved that G is a B+E-group if, and only if, it is a subgroup of a generalized Bassian group.