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

在任意的ABELIAN G组中，如果存在G含有G的纯亚组H的纯亚组h，则在G中可以纯化，该A含有A的a含有A的纯度亚组中包含A的纯子亚组。该亚组H通常称为A的纯船体A。通常，并非所有子组都可以清除。现在我们可以提出以下问题：哪个子组在给定的组中可净化？我们从这个问题开始了这个项目。在这个项目中，我们仅考虑了无扭转的亚组。最后，我们获得了以下结果。（1）我们表征了给出组中无扭转的有限等级可纯化的亚组。（2）我们证明了所有无纯扭转亚组的纯纯素都是同构。我们使用了结果（1）研究分裂问题。分裂问题是表征混合组，这是最大扭转和无扭转亚组的直接总和。如果存在一个纯g的g的纯子k，则我们获得了A…A…A的必要和一个子组A。这样的亚组K称为G中A中的准假船体。我们还可以提出以下问题。在给定的组中，哪个子组可以在给定的组中？对于上述问题，我们在给定的组中表征了无扭转的级别 - 一个准假设亚组。我们使用此结果来显示如何计算直元素的高度纳学。高度纳入很重要。例如，众所周知，当且仅当t（h）与t（k）且H和K的高度 - 静相中是同等的时，无扭转等级1的可计数混合组是同构的。如果G组的亚组a在G中是QuaSi-purifile in Gim-pure，那么最大的QUASI-PURE-PURE-PURE-PURE-PULE-GULS均为G。船体研究最大扭转 - 群的组是扭转的组。这些最大扭转亚组为扭转局部的组包括环状P组的直接产物。这项研究可能是下一个。较少的