具有最小元素同阶类类数的2qp阶群的结构

The Structure of Groups of Order 2qp with the Minimum Class Number of Same Element Order

在有限群中,群的阶和元素的阶对群的结构有很大影响.通过研究群的阶和元素的阶可以得到群的结构、性质,甚至是部分群的分类.群的元素的阶之集所含元素的个数,即同阶类类数,同样对群的结构有很大影响.利用其阶所含素数的个数及群论基础知识,确定了所有阶为2qp的群的同阶类类数的最小值为5,其中q<p是奇素数,并利用数论知识,确定出阶为2qp的同阶类类数为5的群的分类及群的具体结构,详细给出了群的生成元及定义关系.直接利用阶的分类结果,通过计算其元的阶的集合,同样给出了阶为2qp的同阶类类数的最小值为5,再利用阶为2qp的群的分类,从中找出同阶类类数是5的群,其结构与通过理论方法确定出的群的结构是完全一致的.

In finite groups,group order and element order have great influence on the structure of the group.We can get the structure of a group and its property,and even the classification of some groups by researching group order and element order.The number of elements contained in the set of element orders of the group,that is,the number of classes of the same order,also has a great influence on the structure of the group.q<p in groups of order 2qp are odd primes.Using the number of prime numbers contained in its set of orders and the basic knowledge of group theory,it can be determined that the minimum value of the same order class number of all groups with order 2qp is 5.After defining the minimum value of the same order class number of groups with an order of 2qp,the possible situation of the prime graph and the relationship between the prime graph and the structure of the group are determined by using the knowledge of number theory.The classification and structure of group with order 2qp are given.On the other hand,by directly using the classification results of the order,and calculating the set of the order of its elements,the minimum value of the number of the same class with the order of 2qp is also 5.Using the classification of groups with order 2qp,we find out the groups with 5 classes of the same order.The group structure is completely consistent with that determined by theoretical methods.