Quasi-Purifiable Subgroups and Height-Matrices

Abstract

Let G be an arbitrary abelian group. A subgroup A of G is said to be quasi-purifiable in G if a pure subgroup H of G exists containing A such that A is almost-dense in H and H/A is torsion. Such a subgroup H is called a quasi-pure hull of A in G. First we prove that a torsion-free rank-one subgroup A of G is quasi-purifiable in G if and only if, for every prime p and every a ∈ A, h p (a) ≥ ω implies h p (a) = ∞. Next we use the result to compute the height-matrix of the torsion-free element a of an abelian group whose torsion part T(G) is torsion-complete, then all torsion-free rank-one subgroups of G are quasi-purifiable in G and hence the height-matrices of the torsion-free elements of the group G can be computed.