  It is known that a mixed abelian group G with torsion T is Bassian if, and
only if, it has finite torsion-free rank and has finite p-torsion (i.e., each
Tp is finite). 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 with bounded p-torsion (e.g., the balanced-projective groups
and the Warfield groups). We then prove that all generalized Bassian groups
must have finite torsion-free rank, thus answering a question recently posed in
Acta Math. Hung. (2022) by Chekhlov-Danchev-Goldsmith. This implies that every
generalized Bassian group must be a B+E-group; i.e., 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.
