ON m, n-BALANCED PROJECTIVE AND m, n-TOTALLY PROJECTIVE PRIMARY ABELIAN GROUPS Keef, Patrick W.; Danchev, Peter V.;
If and are non-negative integers, then three new classes of abelian -groups are defined and studied: the , -simply presented groups, the , -balanced projective groups and the , -totally projective groups. These notions combine and generalize both the theories of simply presented groups and -projective groups. If , , these all agree with the class of totally projective groups, but when , they also include the -projective groups. These classes are related to the (strongly) n-simply presented and (strongly) -balanced projective groups considered in  and the n-summable groups considered in . The groups in these classes whose lengths are less than are characterized, and if in addition we have , they are determined by isometries of their -socles.