对称群的换位子群

The Commutator Group of Symmetric Group

在一个群中 ,两个换位子的乘积并不一定是换位子。通常对于一个给定的群 ,它的换位子群不能由它的一切换位子所生成的集合而构成 ,只能是由这个集合生成的子群。一个换位子群的所有元素在什么时候都是换位子 ,对于这个问题似乎没有什么好的判定法则或研究结果 ,但在n次对称群中 ,它的换位子群的元素都是换位子 ,即在n次对称群中两个换位子的乘积仍然是一个换位子 ,关于这一结论 ,给出了一种理论证明 ,在此基础上 ,具体给出了一种将n次对称群的换位子群中的元素表示成换位子的方法 ,并利用反例 ,提出存在这样的群G ,它的换位子群中存在这样的元素 。

In a group, the product of two commutators need not be a commutator. Consequently the commutator group of a given group can not be defined as the set of all commutators, but only as the group generated by these. It seems there is no good criteria or investigations on the problem when all elements of the commutator group can be commutators. It shows that in the finite symmetric group S n all elements of its commutator group are commutators in theory. On this basis, the product of two commutators of S n is a commutator. In the following, the representation was given to the elements of commutator group [ S n·S n ]to be expressed by commutators. A counter example was used to show that such a group G existed and its elements contained in commutator group can not be represented by commutators.