|Budget Amount *help
¥2,600,000 (Direct Cost : ¥2,600,000)
Fiscal Year 1999 : ¥1,000,000 (Direct Cost : ¥1,000,000)
Fiscal Year 1998 : ¥1,600,000 (Direct Cost : ¥1,600,000)
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
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.
しかもG/LはTorsion加除群となるので、準基の概念を使ってADE群の分類定理を完成させました。従って、Torsion-Free Rank1のADE群に関しては、構造定理・存在定理、分類定理は完成しました。 Less