ABELIAN GROUPS OF TORSION-FREE RANK 1

首先，我们证明了极大挠子群为挠完全的交换群G的子群在G中是拟纯子群。利用Zom引理，我们可以很容易地看到拟纯子群的极大拟纯壳的存在。我们还证明了群G的拟纯子群的所有极大拟纯壳是同构的。作为这一结果的一个应用，我们证明了对于每个扭转完备群G的子群A，都存在G的一个包含A的极小直和数H，并且这个极小直和数H是G的一个包含A的极小纯纯挠完备子群。当我们将上述结果应用于任意阿贝尔群G时，我们可以证明所有极大扭子群是挠完备的群都是ADE可分解群。将同样的结果应用于扭场秩为1的交换群，我们可以证明：如果X和Y是秩为1的ADE可分解群，则G和H同构当且仅当极大扭子群T(X)和T(Y)同构，高度矩阵H(X)和H(Y)等价.我们可以将扭群的基本子群的概念推广到任意交换群.我们将这些子群命名为“混合基本子群”。首先证明了任意交换群的所有基本子群是同构的，但并不是所有任意交换群的混合基本子群都是同构的；此外，我们证明了1阶自由挠阿贝尔群G存在一个T-高子群L，使得对G的每个T-高阶子群A，(L)型小于或等于(A)型.我们利用这一结果刻画了秩为1的无挠阿贝尔群的所有T-HIH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.