二面体群在一般线性群GL(2,C)中的同构类刻画

On Isomorphic Classes of Dihedral Groups in the General Linear Group GL(2,C)

对两个生成元的非交换有限群在GL（2，C）中的同构矩阵表示以二面体群巩为例进行了探索研究．一方面，利用生成元的特征值为单位根这一特性，采用矩阵对的方法，根据生成元的生成关系对二面体群D。在一般线性群GL（2，C）中的所有同构对象进行了完全刻画．另一方面，对这些同构对象在相似等价的意义下进行了分类，发现这些类的数量为欧拉函数值φ（n）的一半．

In this paper we investigate how a finite non-abelian group of two generators isomorphically represents in GL（2, C） through non-abelian dihedral groups Dn. On one hand, by generating relations of generators of On and the fact that eigenvalues of generators of finite groups are roots of unity, all isomorphic representations of dihedral groups Dn in general linear group GL（2, C） are completely described by using pairs of matrices. On the other hand, all these representations are classified on the condition of similar equivalence, we find the number of these classes is half of the value of euler function φ（n） for n 〉 2.