RESEARCH OF PURIFIABLE AND QUASI-PURIFIABLE SUBGROUPS

在任意交换群G中，如果存在包含A的纯子群H，且该纯子群在包含A的纯子群中极小，则称G的子群A在G中是可纯化的。现在我们可以提出以下问题：在给定的组中，哪个子组是可净化的？我们从这个问题开始这个项目。在这个项目中，我们只考虑了无挠子群。最后，我们得到了如下结果：(1)刻画了给定群中的无挠有限秩纯化子群。(2)证明了可纯化无挠子群的所有纯壳是同构的。利用结果(1)研究了分裂问题。分裂问题是刻划极大挠子群和无挠子群的直和的混合群。得到了有限挠自由秩阿贝尔群是分裂的一个充要条件。a…的子群A更多Belian群G称为在G中拟纯净的，如果存在包含A的纯子群K，使得A在H中几乎稠密，并且H/A是挠的。这样的子群K称为A在G中的拟纯壳。我们还可以提出以下问题：在给定的群中，哪个子群是拟纯化的？在这个项目中，我们刻画了给定群中的无挠秩一拟纯化子群。我们用这个结果来说明如何计算直线元素的高度矩阵。高度矩阵很重要。例如，众所周知，当且仅当T(H)与T(K)同构且H与K的高度矩阵等价时，1阶挠自由的可数混合群H与K同构.如果群G的一个子群A在G中拟纯化，则A在G中存在极大拟纯壳.因此，我们可以利用极大拟纯壳的概念来研究极大扭子群是挠完备的群.极大扭子群是挠完全的群包括循环p-群的直积。这项研究可能是下一个。较少

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