非正规子群共轭类数为4的有限p群的分类

Classification of Finite p-Group With 4 Conjugacy Classes of Non Normal Subgroups

利用非正规子群的共轭类研究有限p群的结构是有限p群领域的前沿问题之一,其中,p表示一个素数。当p>2时,非正规子群的共轭类不超过2p的有限p群已被分类,然而,当p=2时,非正规子群共轭类数为4的有限2群至今未被分类。本文与p>2的分类方法不同,采用中心积和中心扩张的方法对非正规子群共轭类数为4的有限2群进行了研究,分导群的阶为2,4和8三种情况讨论:对于导群的阶为2的群,将其转化为内交换p群的中心积,通过对内交换p群共轭类的讨论,将大部分群转化为内交换p群与循环群的直积;对于导群的阶为4的群,将其商群转化为内交换p群,利用内交换p群的中心扩张进一步转化为一些具体的群;导群的阶为8的群属于广义四元数群。最后通过进一步的讨论,并结合Magma软件的计算,给出了非正规子群共轭类数为4的有限2群的完全分类。

It is one of the frontier problems in the field of finite p-group to study the structure of finite p-group by using the conjugacy classes of nonnormal subgroups,where p represents a prime number.When p is greater than 2,finite p-group whose conjugacy classes of nonnormal subgroups do not exceed 2 p have been classified.However,when p is equal to 2,finite 2-group with 4 conjugacy classes of nonnormal subgroups have not been classified so far.In this paper,different from the classification method of p>2,we used the method of center product and center extension to study the finite 2-group with 4 conjugacy classes of nonnormal subgroups,and discussed three cases where the order of the derived group is equal to 2,4 and 8:For the group with order 2 of derived group,it was transformed into the central product of the minimal non-abelian p-group.By discussing the conjugacy classes of the minimal non-abelian p-group,most groups were transformed into the direct product of the minimal non-abelian p-group and the cyclic group;For the group with order 4 of derived group,its quotient group was transformed into the minimal non-abelian p-group,which was further transformed into some specific groups by using the central extension of the minimal non-abelian p-group;The group with order 8 of the derived group belongs to the generalized quaternion group.Finally,through further discussion and calculation of Magma,the complete classification of finite p-group with 4 conjugacy classes of nonnormal subgroups is given.