权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

STUDY OF ALMOST-DENSE EXTENSION GROUPS

近密扩展群的研究

基本信息

批准号：
10640051
负责人：
OKUYAMA Takashi
金额：
$ 1.66万
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：
1998
资助国家：
日本
起止时间：
1998 至 1999
项目状态：
已结题

项目摘要

Let A be a torsion-free group. An arbitrary abelian group G is said to be an almost-dense extension group (ADE group) of A if A is almost-dense in G and T(G)-high of G. One of the goals of my recent research is to give the structure, the realization, and the classification theorem for ADE groups. L.Fuchs gave an example of the simplest ADE group in his book "Infinite Abelian Groups Vol.2" as Example 2 at p.186. Motivated by this example, I began to study ADE groups.The goal of this project is to study ADE groups of torsion-free rank 1. First, I gave the structure, the realization, and the classification theorem for ADE groups of torsion-free rank 1 whose p-primary subgroup are cyclic for every prime p. An ADE group G is said to be elementary if GィイD2pィエD2 is a direct sum of cyclic group for every prime p. Next, I started studying such elementary ADE groups of torsion-free rank 1. Introducing the concept of quasi-purifiable subgroups into ADE groups of torsion-free rank 1 and defining s … More tandard ADE groups, I established the structure, the realization, and the classification theorem for elementary ADE group of torsion-free rank 1.In general, for every p-group, there exist basic subgroups and all basic subgroups are isomorphic. I extended the concept of basic subgroups from p-group to arbitrary abelian groups. I proved that there exist basic subgroups for every abelian group and all basic subgroups of ADE groups of torsion-free rank 1 are isomorphic. Using basic subgroups, I established the structure theorem. In fact, I proved that an ADE group of torsion-free rank 1 has a moho subgroup and QT-matrices for every prime. Conversely, if there exist a torsion group T, torsion-free rank-one group A, and such matrices for every prime, there exists an ADE group G with T as a maximal torsion subgroup, A as a moho subgroup, and such matrices as QT-matrices. This is the realization theorem.Using the concept of quasi-basis of p-groups, I obtained the classification theorem. It is well-known that the countable mixed groups H and K of torsion-free rank 1 are isomorphic if and only if T(H)ィイD6〜(/)=ィエD6T(K) and the height matrices H(H) and H(K) are equivalent. I proved that the ADE groups L and M of torsion-free rank 1 are isomorphic if and only if T(L)ィイD6〜(/)=ィエD6T(M) and the height matrices H(L) and H(M) are equivalent. Since there exist uncountable ADE groups, I partially deduced this famous result. Less

设A是一个挠自由群。任意交换群G称为A的几乎稠密扩张群(ADE群)，如果A在G中几乎稠密且T(G)-高。我最近研究的目的之一就是给出ADE群的结构、实现和分类定理。L.Fuchs在他的《无限阿贝尔群第二卷》一书中给出了一个最简单的ADE群的例子，作为例子2在第186页。在这个例子的启发下，我开始研究ADE群。本课题的目的是研究秩为1的无挠的ADE群。首先，给出了对每个素数p为循环子群的秩为1的无挠的ADE群的结构、实现和分类定理。如果GィイD2pィエD2是对每个素数p的循环群的直和，则称G是初等的我开始研究这种秩为1的初等无挠ADE群。将拟可纯化子群的概念引入到秩为1的ADE群中，定义了S…在更多的标准ADE群的基础上，建立了无挠秩为1的初等ADE群的结构、实现和分类定理。一般地，对每个p-群，存在基本子群，且所有基本子群同构。我把基本子群的概念从p-群推广到任意交换群。证明了每个交换群都存在基本子群，且1阶自由挠ADE群的所有基本子群同构。利用基本子群，建立了结构定理。事实上，我证明了一个秩为1的无挠的ADE群有一个Moho子群和每个素数的QT-矩阵。反之，如果存在扭群T，无挠一阶群A，且对每个素数都有这样的矩阵，则存在一个ADE群G，其中T是极大扭子群，A是Moho子群，并且有这样的矩阵，如QT-矩阵。利用p-群的拟基的概念，得到了p-群的分类定理。众所周知，1阶挠的可数混合群H和K同构的充要条件是T(H)ィイD6=ィエD6T(K)且高度矩阵H(H)和H(K)等价。证明了1阶无挠ADE群L与M同构的充要条件是T(L)ィイD6=ィエD6T(M)，高度矩阵H(L)与H(M)等价。由于存在不可数的ADE群，我部分地推导了这个著名的结果。较少