CC-子群与有限群结构  

On CC-subgroups and Structure of Finite Groups

作　　者：蔡一[1,2] 刘玲[2] 

机构地区：[1]西南大学数学与统计学院,重庆400715 [2]绵阳东辰国际学校高中部,四川绵阳221000

出　　处：《绵阳师范学院学报》2012年第2期11-12,15,共3页Journal of Mianyang Teachers＇ College

基　　金：国家自然科学基金资助项目(10771172)

摘　　要：该文主要利用CC-子群的存在性来刻画有限群。首先,从CC-子群的存在性推导了一部分已知阶群的结构;其次,推导了当次正规子群和正规子群为CC-子群时的有限群的简单结构,得到了以下主要结论:定理1(1)若|G|=pq,p,q为素数,若G无CC-子群,则G为交换群。(2)若|G|=p2qn,p,q为奇素数,若G的CC-子群个数为1,则G为q幂零群.定理2设G为有限可解群,若G的每个次正规子群均为CC-子群,则|G|=pq。定理3设G为有限可解群,若G的每个正规子群为CC-子群,那么|G|=pqn,G=<a>G',其中,<a>为p阶子群。In this paper,the existence of CC-subgroups is used to describe finite groups.First,the structure of some already-known finite groups is deduced;then,by deducing the simple structure of the finite groups when its subnormal subgroups and normal subgroups are CC-subgroups respectively,the following conclusions are derived： Theorem one（1） if ｜G｜=pq,p,q is prime number and G has no CC-subgroups,hence G is Abel.（2） if ｜G｜=p2qn,p,q is odd prime,and G has only one CC-subgroups and then G is a nilpotent.Theorem two： G is a finite solvable group and every subnormal group is CC-subgroups,p,q are prime then ｜G｜=pq.Theorem three： G is a finite solvable group and every normal subgroups of G are CC-subgroups,then ｜G｜=pqn,G=〈a〉G′,p,q are prime,〈a〉 is a subgroup of order p.

关 键 词：有限群 HALL子群 CC-子群 

分 类 号：O152.1[理学—数学]