Semi-generalized Bassian groups.

As a common non-trivial generalization of the concept of a (proper) generalized Bassian group, we introduce the notion of a <italic>semi-generalized Bassian</italic> group and initiate its comprehensive investigation. Precisely, we give a satisfactory characterization of these groups by showing in the cases of 푝-torsion groups, torsion-free groups and splitting mixed groups their complete description. Moreover, we classify the groups with the clearly related property that every subgroup is essential in a direct summand of the whole group. [ABSTRACT FROM AUTHOR]