某些素数幂阶次正规子群与有限群的p-幂零性
发布时间：2018-10-13 12:39
【摘要】：子群的一些性质对群的结构往往有着很大的影响,利用有限群的某些子群的性质来研究有限群的结构是目前许多群论研究者常用的方法.H.Wielandt于1939年提出了有限群的次正规子群这一概念.设G是一个有限群,称H为G的次正规子群,如果存在G的某个次正规群列H=H0H1(?)…(?)Hn = G.次正规子群对有限群的影响已有许多研究,本文将一些与次正规子群有关的结果进行推广,得到了有限群G的p-幂零群的若干新的条件.本文按照内容共分为两章:第一章主要是介绍一些相关定义与概念,以及利用有限群的某些子群的特性研究有限群结构的研究背景和本文所需的基本结果和相关引理.第二章在第一章的基础上分别利用Sylowp p-群的极大子群和2-极大子群的次正规性得到了群G是p-幂零的若干充分条件.主要结果如下:定理2.1.1设G是有限群,p是|G|的素因子,P是G的一个Sylow p-子群,若P的每个非循环极大子群次正规于G,且NG(P)是p-幂零的,则G是p-幂零的.定理2.1.6设G是有限群,p是|G|的素因子,P是G的一个Sylow p-子群,若P的每个极大子群P1次正规于G,且NG(P1)是 p-幂零的,且G是M(pn,q)-无关的,则G是p-幂零的.定理2.1.10设G是有限群,p是|G|的素因子,群G的Sylow p-子群P的每个极大子群P1次正规于G,且(|G|,P-1)= 1.G是A4-无关的,则G是p-幂零的.定理2.2.1设G是有限群,p是|G|的最小素因子,P是G的一个Sylow p-子群,若P的每个2-极大子群P2次正规于G,且NG(P2)是p-幂零的,则G是p-幂零的.定理2.2.5设G是有限群,p是|G|的最小素因子,P是G的一个Sylow p-子群,若P的每个2-极大子群P2次正规于G.且NG(P)是p-幂零的,则G是p-幂零的.定理2.2.10设G是有限群,p是|G|的最小素因子,群G的Sylow p-子群P的每个2-极大子群P2次正规于G.且(|G|,p-1)= 1.且G是A4-无关的.则G是p-幂零的.
[Abstract]:Some properties of subgroups often have a great influence on the structure of groups. Using the properties of some subgroups of finite groups to study the structure of finite groups is a method commonly used by many group theorists at present. H.Wielandt put forward the concept of subnormal subgroups of finite groups in 1939. Let G be a finite group and H be a subnormal subgroup of G. if there exists a sequence of subnormal groups of G H=H0H1 (?). (?) Hn = G. There are many studies on the influence of subnormal subgroups on finite groups. In this paper, we generalize some results related to subnormal subgroups and obtain some new conditions for pnilpotent groups of finite groups G. This paper is divided into two chapters according to the content: the first chapter mainly introduces some related definitions and concepts, and studies the structure of finite groups by using the properties of some subgroups of finite groups, and the basic results and relevant Lemma needed in this paper. In chapter 2, by using the subnormality of maximal subgroups and 2-maximal subgroups of Sylowp p-groups, we obtain some sufficient conditions for G to be p-nilpotent. The main results are as follows: theorem 2.1.1 Let G be a finite group, p be a prime factor of G, P be a Sylow p-subgroup of G. if every noncyclic maximal subgroup of P is subnormal to G and NG (P) is pnilpotent, then G is pnilpotent. Theorem 2.1.6 Let G be a finite group, p be a prime factor of G, P be a Sylow p- subgroup of G. if every maximal subgroup P1 of P is normal to G, and NG (P1) is pnilpotent and G is M (pn,q) -independent, then G is pnilpotent. Theorem 2.1.10 Let G be a finite group, p be a prime factor of G, every maximal subgroup P1 of Sylow p- subgroup P of G be normal to G, and (G, P-1) = 1. G is A4-independent, then G is pnilpotent. Theorem 2.2.1 Let G be a finite group, p be the least prime factor of G, P be a Sylow p-subgroup of G. if every 2-maximal subgroup of P is normal to G, and NG (P2) is pnilpotent, then G is pnilpotent. Theorem 2.2.5 Let G be a finite group, p be the least prime factor of G, P be a Sylow p- subgroup of G. if every 2-maximal subgroup of P is normal to G. And if NG (P) is pnilpotent, then G is pnilpotent. Theorem 2.2.10 Let G be a finite group, p be the least prime factor of G, and every 2-maximal subgroup P2 of Sylow p- subgroup P of group G be normal to G. And (G, p-1) = 1. And G is A4-independent. Then G is p-nilpotent.
【学位授予单位】：广西师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O152.1
本文编号：2268665
[Abstract]:Some properties of subgroups often have a great influence on the structure of groups. Using the properties of some subgroups of finite groups to study the structure of finite groups is a method commonly used by many group theorists at present. H.Wielandt put forward the concept of subnormal subgroups of finite groups in 1939. Let G be a finite group and H be a subnormal subgroup of G. if there exists a sequence of subnormal groups of G H=H0H1 (?). (?) Hn = G. There are many studies on the influence of subnormal subgroups on finite groups. In this paper, we generalize some results related to subnormal subgroups and obtain some new conditions for pnilpotent groups of finite groups G. This paper is divided into two chapters according to the content: the first chapter mainly introduces some related definitions and concepts, and studies the structure of finite groups by using the properties of some subgroups of finite groups, and the basic results and relevant Lemma needed in this paper. In chapter 2, by using the subnormality of maximal subgroups and 2-maximal subgroups of Sylowp p-groups, we obtain some sufficient conditions for G to be p-nilpotent. The main results are as follows: theorem 2.1.1 Let G be a finite group, p be a prime factor of G, P be a Sylow p-subgroup of G. if every noncyclic maximal subgroup of P is subnormal to G and NG (P) is pnilpotent, then G is pnilpotent. Theorem 2.1.6 Let G be a finite group, p be a prime factor of G, P be a Sylow p- subgroup of G. if every maximal subgroup P1 of P is normal to G, and NG (P1) is pnilpotent and G is M (pn,q) -independent, then G is pnilpotent. Theorem 2.1.10 Let G be a finite group, p be a prime factor of G, every maximal subgroup P1 of Sylow p- subgroup P of G be normal to G, and (G, P-1) = 1. G is A4-independent, then G is pnilpotent. Theorem 2.2.1 Let G be a finite group, p be the least prime factor of G, P be a Sylow p-subgroup of G. if every 2-maximal subgroup of P is normal to G, and NG (P2) is pnilpotent, then G is pnilpotent. Theorem 2.2.5 Let G be a finite group, p be the least prime factor of G, P be a Sylow p- subgroup of G. if every 2-maximal subgroup of P is normal to G. And if NG (P) is pnilpotent, then G is pnilpotent. Theorem 2.2.10 Let G be a finite group, p be the least prime factor of G, and every 2-maximal subgroup P2 of Sylow p- subgroup P of group G be normal to G. And (G, p-1) = 1. And G is A4-independent. Then G is p-nilpotent.
【学位授予单位】：广西师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O152.1
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