Direct Sums of Quotient Divisible Groups

Abstract

An abelian group G is quotient divisible (qd) if it is of finite torsion-free rank and there exists a free subgroup F ⊂ G with G/F a divisible torsion group. We study the category ∑𝒟 with objects arbitrary direct sums of qd groups and maps quasi-homomorphisms. First we identify the qd-indecomposable groups, that is the indecomposables in the category ∑𝒟. These turn out to be the qd groups G such that, if G ⊃ A ⊕ B ⊃ tG for t > 0, then A or B is finite. The major tool for studying ∑𝒟 is the notion of an admissable quasi-decomposition G ≈ ⊕ G _k of a group G ∈ ∑𝒢 into a direct sum of qd-indecomposable subgroups. (See Definition 6 below.) For any qd-indecomposable group A and any admissable quasi-decomposition G ≈ ⊕ G _k , the number of G _k quasi-isomorphic to A is an invariant of G, denoted σ _A (G). The cardinal numbers σ _A (G), together with the finite Ulm invariants (G), form a complete set of isomorphism invariants for groups G = ⊕G _i  ∈ ∑𝒢 with p-rank[G _i /T(G _i )] ≤ 1 for all p, i. Let G ∈ ∑𝒟 with G = A ⊕ B. If G ≈ ⊕ G _k , then there is an admissable quasi-decomposition A ≈ ⊕ A _j such that each A _j is quasi-isomorphic to some G _k(j). We consider the problem: When is an abelian group A with an admissable quasi-decomposition A ≈ ⊕ A _j into qd-indecomposable subgroups a direct summand of a group in ∑𝒟? We get a positive answer if A is torsion-free or if the mixed groups A _j are all in a class 𝒢 ⊂ 𝒟.

Keywords:

• Mixed abelian groups
• Quasi-decompositions

Acknowledgment

I wish to thank the referee for his/her careful reading of the paper and helpful suggestions for improvement in the exposition.