On m,n-balanced projective and m,n-totally projective primary abelian groups

If m and n are non-negative integers, then three new classesof abelian p-groups are defined and studied: the m, n-simply presentedgroups, the m, n-balanced projective groups and the m, n-totally projec-tive groups. These notions combine and generalize both the theories ofsimply presented groups and pw+n-projective groups. If m, n = 0, theseall agree with the class of totally projective groups, but when m+n≥1,they also include the pw+m+n-projective groups. These classes are relatedto the (strongly) n-simply presented and (strongly) n-balanced projectivegroups considered in [15] and the n-summable groups considered in [2]. The groups in these classes whose lengths are less than w2 are characterized, and if in addition we have n = 0, they are determined by isometriesof their pm-socles.