正规子群求解方法的一个注记

A Note on the Normal Subgroups

Cayley定理是抽象代数中一个非常重要的定理.因为这个定理建立了抽象的有限群G和一个具体群S_n之间的联系.即G同构于S_n的一个子群.所以,对于S_n的子群的研究就显得尤其重要.但是,在教学实践中,学生只是通过定义来求S_n或是S_n的子群的正规子群往往是很困难的事情.本文给出了在群论和表示论中经常用到求S_n的正规子群的一种方法.通过这种方法,希望可以加深学生对相应知识的理解.

The Cayley theorem plays an important role in modern algebras.Because it establishes a relation between a general abstract group G and a symmetric group Sn.Namely,G is isomorphic to a subgroup of Sn.So it is meaningful to study the subgroups of symmetric groups carefully.However,it is difficult for students to determine the normal subgroups precisely for the heavy computation.In this note,we give a method which occurs frequently in group theory and representation theory to decide the normal subgroups of symmetric groups.We hope that it will be useful for the beginners.