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 [15] and the n-summable groups considered in [2]. 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.