中心循环的亚循环p-群

Metacyclic p-group of Central Cycle

给出了亚循环p-群G的中心循环的条件,当p为奇素数时,中心循环的充要条件是u=r;当G是I型2-群时,中心循环充要条件是u=r;当G是Ⅱ型2-群时,中心循环充要条件是u=0且t′=r或者u=1。另外,得到了中心循环的亚循环p-群都不是p-中心p-群;对于亚循环p-群G,p为奇素数,当u<r,G是p-中心p-群。

In this paper, the conditions for the central cycle of metacyclic p-groups G are given. When p is an odd prime number, the necessary and sufficient condentrition of central cycle is u=r. When G is type I of 2-groups, the necessary and sufficient condition of central cycle is u=r.When G is type Ⅱ of 2-groups, the necessary and sufficient conditions for central cycle are u=0 and t′=r or u=1. In addition, it is obtained that the metacyclic groups of central cycle are not p-central p-groups. For metacyclic p-groups with p is an odd prime number, when u<r, G is a p-central p-group.