幂零类是2的有限幂零群的轨道长度  

Orbit Sizes of Finite Nilpotent Groups of Class 2

作　　者：薛海波[1] 吕恒[2] XUE Haibo;LU Heng(School of Mechatronics and Information Engineering,Chongqing College of Humanities,Science and Technology,Hechuan Chongqing 401524;School of Mathematics and Suatistics.Southwest University,Chongqing 400715,China)

机构地区：[1]重庆人文科技学院机电与信息工程学院,重庆合川401524 [2]西南大学数学与统计学院,重庆400715

出　　处：《重庆师范大学学报（自然科学版）》2023年第5期99-102,共4页Journal of Chongqing Normal University:Natural Science

基　　金：国家自然科学基金面上项目(No.11971391)。

摘　　要：为研究有限幂零群G忠实作用在一个可解群H上的轨道长度,假设有限幂零群G忠实不可约作用在一个初等交换q-群V上,则可得Z(G)是循环群,且对任意V中元v,中心化子CG(v)与Z(G)交一定等于1,考虑中心化子阶的情况。假设G是幂零类为2的有限群且Z(G)是循环群,若子群S满足|S|^(2)>|G|,则S与中心Z(G)交不等于1。若G忠实不可约作用在初等交换q-群V上,证明了所有的最小轨道长度的平方大于等于群G的阶。If a nilpotent group G acts faithfully on a solvable group H,it turned out to be helpful to know the orbit sizes of H in this action.Suppose that a nilpotent group G acts faithfully and irreducibly on V.It is well known that Z(G)is cyclic and the intersection of C_G(v)and Z(G)equals to 1 for any nontrivial element v in V.Let G be a nilpotent group of class 2 with Z(G)cyclic.If S is a subgroup of G with|S|^2>|G|,then the intersection of S and Z(G)is not trival.If G acts faithfully and irreducibly on an elementary abelian N,then the minimal orbit has size large than|G|^(1/2).

关 键 词：有限P-群 轨道 幂零类 

分 类 号：O152[理学—数学]