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RESEARCH OF PURIFIABLE AND QUASI-PURIFIABLE SUBGROUPS

可纯化和准可纯化子群的研究

基本信息

批准号：
13640053
负责人：
OKUYAMA Takashi
金额：
$ 0.96万
依托单位：
TOBA NATIONAL COLLEGE OF MARITIME TECHNOLOGY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640053/
关键词：
Purifiable Subgroups Quasi-Purifiable Subgroups p-overhang set The maximal torsion subgroup Torsion-Free subgroup T-High Subgroup Height-Matrices Torsion-Free Rank 準純粋包をもつ部分群純粋包 p-overhang set 準純粋包 empty-p-overhang set部分群最大ねじれ部

项目摘要

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

在任意阿贝尔群G中，称G的子群A在G中可纯化，如果存在G的包含A的纯子群H，且H是G的包含A的纯子群中最小的。这样的子群H称为A的纯船体。一般来说，并非所有子组都是可纯化的。现在我们可以提出以下问题：在给定的群中，哪个子群是可净化的？我们从这个问题开始了这个项目。在这个项目中，我们只考虑无挠子群。最后，我们得到了以下结果。(1)刻划了给定群中的无挠有限秩可纯化子群。(2)证明了可纯挠自由子群的纯壳同构，并利用结果（1）研究了分裂问题。分裂问题是刻画由极大挠子群与无挠子群的直和构成的混合群。本文给出了有限挠自由秩的阿贝尔群是可裂群的一个充要条件。 ...更多信息称Belian群G在G中是拟可纯化的，如果存在G的纯子群K包含A，使得A在H中几乎稠密，且H/A是挠的.这样的子群K称为A在G中的拟纯船体。我们也可以提出下面的问题：在一个给定的群中，哪个子群是拟纯化的？在本项目中，针对上述问题，我们刻画了给定群中的无挠秩一拟可纯化子群。我们用这个结果来说明如何计算直元素的高度矩阵。高度矩阵很重要。例如，已知无挠秩为1的可数混合群H与K同构当且仅当T（H）与T（K）同构且H与K的高度矩阵等价，若群G的子群A在G中拟纯，则存在A在G中的极大拟纯船体.因此，我们可以利用极大拟纯壳的概念来研究极大挠子群是挠完备的群。这些群的最大挠子群是挠完全的，包括循环p-群的直积。这项研究可能是下一个。少