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whose seminal work on the homological aspects of abelian group theory continues to inspire the authors Abstract. If m and n are non-negative integers, then three new classes of abelian p-groups are defined and studied: the m, n-simply presented groups, the m, n-balanced projective groups and the m, n-totally projective groups. These notions combine and generalize both the theories of simply presented groups and p ω+n -projective groups. If m, n = 0, these all agree with the class of totally projective groups, but when m + n ≥ 1, they also include the p ω+m+n -projective groups. These classes are related to the (strongly) n-simply presented and (strongly) n-balanced projective groups considered in [15] and the n-summable groups considered in [2]. The groups in these classes whose lengths are less than ω 2 are characterized, and if in addition we have n = 0, they are determined by isometries of their p m -socles.

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We present a number of generalizations of a classical result of Wallace regarding countable extensions of totally projective primary abelian groups.

Several equivalent descriptions are given of the class of primary abelian groups whose separable subgroups are all direct sums of cyclic groups; such groups are called ω-totally Σ-cyclic. This establishes the converse of a theorem due to Megibben. For n < ω, this is generalized to a consideration of the class of primary abelian groups whose p ω+n -bounded subgroups are all p ω+n -projective. The question of whether there are such groups that are proper in the sense that they are neither p ω+n -projective nor ω-totally Σ-cyclic is shown to be logically equivalent to a natural question about the structure of valuated vector spaces. Finally, it is shown that both of these statements are independent of ZFC.Mathematics Subject Classification (2010). Primary 20K10.