有限生成Abelian群同构类的计算

The Calculation of the Isomorphism Classes of a Finitely Generated Abelian Group

通过用生成元及其满足的关系来定义有限生成Abelian群,并且用一个矩阵表示此关系,然后通过理论分析将有限生成Abelian群同构类这一问题转化为将此矩阵通过初等变换化为满足一定条件的对角矩阵的问题,进而求出有限生成Abelian群的结构。最后,我们把该方法在计算机上实现,当矩阵较大时,我们将使计算更加容易和便利。

In this paper, we give a method to calculate the isomorphism classes of a finitely generated Abelian group, by using the generators and the relation of the generators to define the finitely generated Abelian group and using a matrix to represent the relation. Through the theoretical analysis, the problem about the isomorphism classes of a finitely generated Abelian group is translated into the problem about the diagonal matrix of the relation matrix. So that one could determine easily the isomorphism classes of any finitely gener- ated Abelian group once its generators and the relation are given. In the last, we also implement the method on the computer, which will make the computation much easier, especially when the matrix gets bigger.