ABELIAN GROUPS OF TORSION-FREE RANK 1

首先证明了最大扭转子群为扭转完全的阿贝尔群G的所有子群在G中都是拟可纯化的，利用Zom引理可以很容易地看出拟可纯化子群的极大拟纯壳的存在。证明了群G的拟可净化子群的所有极大拟纯壳是同构的。作为这一结果的应用，我们证明了对于扭转完全群G的每一个子群A，存在一个包含A的G的最小直接和H，并且这个最小直接和H是包含A的G的最小纯扭转完全子群。当我们将上述结果应用于任意阿贝群G时，我们可以证明所有最大扭转子群为扭转完全的群都是ADE可分解群。将同样的结果应用到无扭转秩为1的阿贝尔群上，证明了如果X和Y是无扭转秩为1的ADE可分解群，则当且仅当最大扭转子群T(X)和T(Y)同构且高度矩阵H(X)和H(Y)相等时，G和H是同构的。我们可以将扭转群的基本子群的概念推广到任意阿贝尔群。我们将这些子群命名为“混合基本子群”。首先，我们证明了并非所有的任意阿贝尔群的混合基本子群都是同构的，尽管所有的扭转群的基本子群都是同构的。此外，我们证明了无扭转秩为1的阿贝尔群G存在一个t -高子群L，使得G的每一个t -高子群a的型(L)小于或等于型(a)，并利用这一结果刻画了无扭转秩为1的阿贝尔群的所有t -高子群同构。

First we proved that all subgroups of abelian groups G whose maximal torsion subgroups are torsion-complete are quasi-purifiable in G. We can easily see that there exist maximal quasi-pure hulls of quasi-purifiable subgroups by Zom's Lemma. We also proved that all maximal quasi-pure hulls of quasi-purifiable subgroups of the groups G are isomorphic. As an application of this result, we proved that, for every subgroup A of torsion-complete groups G, there exists a minimal direct summand H of G containing A and such a minimal direct summand H is a minimal pure torsion-complete subgroup of G containing A. When we apply the above result to arbitrary abelian group G, we can show that all groups whose maximal torsion subgroups are torsion-complete are ADE decomposable groups. Applying the same result to abelian groups of torsion-fiee rank 1, we can prove that if X and Y are ADE decomposable groups of torsion-free rank 1, then G and H are isomorphic if and only if the maximal torsion subgroups T(X) and T(Y) are isomorphic and the height-matrices H(X) and H(Y) are equivalent.We could extend the concept of basic subgroups of torsion groups to arbitrary abelian groups. We named the subgroups "mixed basic subgroups". First, we showed that not all mixed basic subgroups of arbitray abelian groups are isomorphic, though all basic subgroups of torsion groups are isomorphic.Moreover, we proved that there exists a T-high subgroup L of an abelian group G of torsion-free rank 1 such that type(L) is less than or equal to type(A) for every T-high sugroup A of G. We used this result to characterize the abelian group of torsion-free rank 1 all of whose T-hih subgroups are isomorphic.