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)

Research Abstract 
Let A be a torsionfree group. An arbitrary abelian group G is said to be an almostdense extension group (ADE group) of A if A is almostdense 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 torsionfree rank 1. First, I gave the structure, the realization, and the classification theorem for ADE groups of torsionfree rank 1 whose pprimary 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 torsionfree rank 1. Introducing the concept of quasipurifiable subgroups into ADE groups of torsionfree rank 1 and defining s
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tandard ADE groups, I established the structure, the realization, and the classification theorem for elementary ADE group of torsionfree rank 1. In general, for every pgroup, there exist basic subgroups and all basic subgroups are isomorphic. I extended the concept of basic subgroups from pgroup to arbitrary abelian groups. I proved that there exist basic subgroups for every abelian group and all basic subgroups of ADE groups of torsionfree rank 1 are isomorphic. Using basic subgroups, I established the structure theorem. In fact, I proved that an ADE group of torsionfree rank 1 has a moho subgroup and QTmatrices for every prime. Conversely, if there exist a torsion group T, torsionfree rankone 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 QTmatrices. This is the realization theorem. Using the concept of quasibasis of pgroups, I obtained the classification theorem. It is wellknown that the countable mixed groups H and K of torsionfree 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 torsionfree 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
