| CC-子群与有限群结构 |
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| 引用本文： | 蔡一,刘玲.CC-子群与有限群结构[J].绵阳师范学院学报,2012,31(2):11-12,15. |
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| 作者姓名： | 蔡一 刘玲 |
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| 作者单位： | 1. 西南大学数学与统计学院,重庆400715／绵阳东辰国际学校高中部,四川绵阳221000
2. 绵阳东辰国际学校高中部,四川绵阳,221000 |
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| 基金项目： | 国家自然科学基金资助项目 |
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| 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。
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| 关 键 词： | 有限群 Hall子群 CC-子群 |
On CC-subgroups and Structure of Finite Groups |
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| CAI Yi,LIU Ling.On CC-subgroups and Structure of Finite Groups[J].Journal of Mianyang Normal University,2012,31(2):11-12,15. |
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| Authors: | CAI Yi LIU Ling |
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| Institution: | 1.School of Mathematics & Statistics,Southwest University,Chongqing 400715; 2.Dept.of Senior High School,Mianyang Dongchen International School,Mianyang Sichuan 621000) |
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| Abstract: | 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. |
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| Keywords: | Finite groups Hall subgroup CC-subgroups |
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