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，如果存在包含 A 的 G 的纯子群 H，且该纯子群 H 在包含 A 的 G 的纯子群中最小，则称 G 的子群 A 在 G 中是可纯化的。这样的子群 H 称为一般而言，并非所有子群都是可纯化的。现在我们可以提出以下问题：在给定群中哪个子群是可纯化的？我们从这个问题开始这个项目，我们只考虑了无扭转。最后，我们得到了以下结果。（1）我们刻画了给定群中的无挠有限秩可纯化子群。（2）我们证明了所有可纯化无挠子群的纯壳都是同构的。我们使用结果（1））来研究分裂问题。分裂问题是表征最大扭转子群和无扭转子群的直和的混合群。我们得到了阿贝尔群的充要条件。如果存在包含 A 的 G 的纯子群 K，使得 A 在 H 中几乎稠密，并且H/A 是挠率。这样的子群 K 称为 G 中 A 的拟纯壳。我们还可以提出以下问题。在给定群中哪个子群是可拟纯的？在本项目中，对于上述问题问题中，我们描述了给定组中的无扭转秩一准可纯化子群，我们使用这个结果来展示如何计算直元素的高度矩阵，例如，众所周知。无扭转秩 1 的可数混合群 H 和 K 同构当且仅当 T(H) 同构于 T(K) 并且 H 和 K 的高度矩阵相等。如果一个子群群G的A在G中是准可纯的，那么G中存在A的最大拟纯壳。因此，我们可以使用最大拟纯壳的概念来研究最大挠子群为挠子群的群。这些最大挠率子群是挠率完全的群包括循环 p 群的直接乘积。