Let G be an abelian p-group for some prime p. A subgroup A of G is said to be purifiable if the set of all pure subgroups of G containing A contains a minimal element. Any minimal element in this set is said to be a pure hull of A in G. The groups for which every subgroup is purifiable are precisely the ones that are bounded modulo their maximal divisible subgroups. The authors consider the problem of characterizing the purifiable subgroups of a given p-group G. As a tool, they introduce the nth-overhang \(V_ n(G,A)\) of A in G which is defined as the kernel of the natural map from the n-th Ulm-Kaplansky invariant \(U_ n(G,A[p])\) of G relative to A[p] onto \(U_ n(G,A)\). It is shown that, if A is purifiable, \(V_ n(G,A)\) is zero for all but finitely many n. The converse is false but holds in case A is almost-dense in G. Purifiability is related to that of basic subgroups: A is purifiable iff all of its basic subgroups have this property; and, while two pure hulls of a purifiable subgroup A need not be isomorphic, their basic subgroups are.