On the complexity of the classification problem for torsion-free abelian groups of finite rank
On the complexity of the classification problem for torsion-free abelian groups of finite rank
复制标题
有限秩无扭转阿贝尔群分类问题的复杂性
作者：
S. Thomas
在本文中，我们将讨论最近的一些贡献的项目[15，14，2，18，22，23]的解释，为什么还没有令人满意的系统的完全不变量的有限秩n ≥ 2的挠自由阿贝尔群。回想一下，直到同构，秩为n的无挠阿贝尔群恰好是n维向量空间n的加法子群，其中包含n个线性无关元素。因此，秩不超过n的无挠阿贝尔群的集合可以自然地等同于由n的所有非平凡加法子群组成的集合S（n）。在1937年，Baer [4]解决了秩为1的群的类S（n）的分类问题如下。设f为素数的集合。设G是无挠阿贝尔群，0 n {∞}; x的特征χ（x）定义为函数
In this paper, we shall discuss some recent contributions to the project [15, 14, 2, 18, 22, 23] of explaining why no satisfactory system of complete invariants has yet been found for the torsion-free abelian groups of finite rank n ≥ 2. Recall that, up to isomorphism, the torsion-free abelian groups of rank n are exactly the additive subgroups of the n -dimensional vector space ℚ n which contain n linearly independent elements. Thus the collection of torsion-free abelian groups of rank at most n can be naturally identified with the set S (ℚ n ) of all nontrivial additive subgroups of ℚ n . In 1937, Baer [4] solved the classification problem for the class S (ℚ)of rank 1 groups as follows. Let ℙ be the set of primes. If G is a torsion-free abelian group and 0 ≠ x ϵ G , then the p-height of x is defined to be h x ( p ) = sup{ n ϵ ℕ ∣ There exists y ϵ G such that p n y = x } ϵ ℕ ∪{∞}; and the characteristic χ (x) of x is defined to be the function