ABELIAN GROUPS OF TORSION-FREE RANK 1

首先证明了交换群G的所有极大挠子群是挠完备的子群在G中是拟纯化的。利用Zom引理，我们可以很容易地看出，存在拟可纯化子群的极大拟纯壳。证明了群G的拟可纯化子群的极大拟纯壳同构。作为这一结果的应用，我们证明了：对于扭完备群G的每个子群A，存在G的包含A的极小直和项H，并且该极小直和项H是G的包含A的极小纯扭完备子群。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挠子群是挠完备的群都是ADE可分解群。将同样的结果应用于挠自由秩为1的阿贝尔群，证明了：若X和Y是挠自由秩为1的ADE可分解群，则G和H同构当且仅当极大挠子群T（X）和T（Y）同构，且高矩阵H（X）和H（Y）等价，从而将挠群的基本子群的概念推广到任意阿贝尔群.我们把这类子群命名为“混合基本子群”。首先证明了任意阿贝尔群的混合基本子群不是同构的，而扭群的基本子群是同构的，并证明了无扭秩为1的阿贝尔群G存在一个T-高子群L，使得对G的每个T-高子群A，（L）型小于或等于（A）型.我们利用这个结果刻画了无挠秩为1的交换群的所有T-hih子群是同构的。

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.