用交换子群的个数刻画A5和S5

Characterizing A5 and S5 by the Numbers of Its Abelian Subgroups

利用完全初等的方法刻画交错群A5和对称群S5.得到如下结论:若群G的阶为60,则G≌A5的充要条件是G的交换子群的阶为2,3,4,5,且个数分别是15,10,5,6;若群G的阶为120,则G≌S5的充要条件是G的交换子群的阶为2,3,4,5,6,且个数分别是25,10,35,6,10.

It is proved that A5 and S5 can be uniquely determined by its order and the numbers of its abelian subgroups of different orders by elementary approaches.The conclusions are as following:If the order of group G is 60,then G≌A5 if and only if the orders of the abelian subgroups of G are 2,3,4,5 and the numbers of them are 15,10,5,6 respectively;If the order of group G is 120,then G≌S5 if and only if the orders of the abelian subgroups of G are 2,3,4,5,6,and the numbers of them are 25,10,35,6,10 respectively.