RESEARCH OF PURIFIABLE AND QUASI-PURIFIABLE SUBGROUPS

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In an arbitrary abelian group G, a subgroup A of G is said to be purifiable in G if there exists a pure subgroup H of G containing A which is minimal among the pure subgroups of G that contain A. Such a subgroup H is called a pure hull of A. In general, not all subgroups are purifiable. Now we can pose the following problem: Which subgroup is purifiable in a given group?We started this project with the problem. In this project, we considered only torsion-free subgroups. Finally, we obtained the following results.(1) We characterized torsion-free finite rank purifiable subgroups in give groups.(2) We proved that all pure hulls of purifiable torsion-free subgroups are isomorphic.We used the result (1) to study the splitting problem. The splitting problem is to characterize mixed groups which are a direct sum of the maximal torsion and torsion-free subgroup. We obtained a necessary and sufficient condition for abelian groups of finite torsion-free rank to be splitting.A subgroup A of an a … More belian group G is said to be quasi-purifiable in G if there exists a pure subgroup K of G containing A such that A is almost-dense in H and H/A is torsion. Such a subgroup K is called a quasi-pure hull of A in G. We can also pose the following problem.Which subgroup is quasi-purifiable in a given group?In this project, for the above problem, we characterized torsion-free rank-one quasi-purifiable subgroups in given groups. We used this result to show how to calculate the height-matrices of straight elements. Height-matrices are important. For example, it is well-known that countable mixed groups H and K of torsion-free rank 1 are isomorphic if and only if T(H) is isomorphic to T(K) and the height-matrices of H and K are equivalent.If a subgroup A of a group G is quasi-purifiable in G, then there exists a maximal quasi-pure hull of A in G. So we might use the concept of maximal quasi-pure hulls to study the groups whose maximal torsion-subgroups are torsion-complete. These groups whose maximal torsion subgroups are torsion-complete include direct products of cyclic p-groups. This study could be next. Less