某些素?cái)?shù)冪階次正規(guī)子群與有限群的p-冪零性
發(fā)布時(shí)間：2018-10-13 12:39
【摘要】：子群的一些性質(zhì)對(duì)群的結(jié)構(gòu)往往有著很大的影響,利用有限群的某些子群的性質(zhì)來(lái)研究有限群的結(jié)構(gòu)是目前許多群論研究者常用的方法.H.Wielandt于1939年提出了有限群的次正規(guī)子群這一概念.設(shè)G是一個(gè)有限群,稱H為G的次正規(guī)子群,如果存在G的某個(gè)次正規(guī)群列H=H0H1(?)…(?)Hn = G.次正規(guī)子群對(duì)有限群的影響已有許多研究,本文將一些與次正規(guī)子群有關(guān)的結(jié)果進(jìn)行推廣,得到了有限群G的p-冪零群的若干新的條件.本文按照內(nèi)容共分為兩章:第一章主要是介紹一些相關(guān)定義與概念,以及利用有限群的某些子群的特性研究有限群結(jié)構(gòu)的研究背景和本文所需的基本結(jié)果和相關(guān)引理.第二章在第一章的基礎(chǔ)上分別利用Sylowp p-群的極大子群和2-極大子群的次正規(guī)性得到了群G是p-冪零的若干充分條件.主要結(jié)果如下:定理2.1.1設(shè)G是有限群,p是|G|的素因子,P是G的一個(gè)Sylow p-子群,若P的每個(gè)非循環(huán)極大子群次正規(guī)于G,且NG(P)是p-冪零的,則G是p-冪零的.定理2.1.6設(shè)G是有限群,p是|G|的素因子,P是G的一個(gè)Sylow p-子群,若P的每個(gè)極大子群P1次正規(guī)于G,且NG(P1)是 p-冪零的,且G是M(pn,q)-無(wú)關(guān)的,則G是p-冪零的.定理2.1.10設(shè)G是有限群,p是|G|的素因子,群G的Sylow p-子群P的每個(gè)極大子群P1次正規(guī)于G,且(|G|,P-1)= 1.G是A4-無(wú)關(guān)的,則G是p-冪零的.定理2.2.1設(shè)G是有限群,p是|G|的最小素因子,P是G的一個(gè)Sylow p-子群,若P的每個(gè)2-極大子群P2次正規(guī)于G,且NG(P2)是p-冪零的,則G是p-冪零的.定理2.2.5設(shè)G是有限群,p是|G|的最小素因子,P是G的一個(gè)Sylow p-子群,若P的每個(gè)2-極大子群P2次正規(guī)于G.且NG(P)是p-冪零的,則G是p-冪零的.定理2.2.10設(shè)G是有限群,p是|G|的最小素因子,群G的Sylow p-子群P的每個(gè)2-極大子群P2次正規(guī)于G.且(|G|,p-1)= 1.且G是A4-無(wú)關(guān)的.則G是p-冪零的.
[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.
【學(xué)位授予單位】：廣西師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O152.1
本文編號(hào)：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.
【學(xué)位授予單位】：廣西師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O152.1
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