子群的正规化子与有限p-群的结构

发布时间：2018-05-21 21:27

本文选题：有限p-群 + 正规化子　；参考：《上海大学》2016年博士论文

【摘要】：在有限群论的研究中,子群的正规性与子群间的某种交换性是人们研究的基本出发点,而子群的正规化子与中心化子是子群的正规性和交换性的一种度量.于是利用子群的正规化子和中心化子来研究有限群的结构成为人们非常感兴趣的研究课题,并且获得了大量的研究成果.本文也将从子群的正规化子和中心化子出发来研究有限p-群的结构,同时我们对秩为2的有限2-群也做了进一步的研究.第三章研究了2平衡p-群的结构.称有限群G为n平衡群,若对G中任意满足d(H)≥n,d(K)≥n的子群H,K,有H≤NG(K)或K≤NG(H).我们不仅完全分类了二元生成2平衡p-群,而且证明了三元生成2平衡p-群G中存在正规的亚循环子群N使得G/N循环.对于生成元个数大于等于4的2平衡p-群G,我们还证明了G为模群,且G'≤(x),其中x是G中满足o(x)=exp(G)的任意元素.第四章研究了CAC-p-群的结构.称有限p-群G为CAC-p-群,如果对任意不包含在G的中心的非循环交换子群H有CG(H)/H循环.本章给出了CAC-p-群的完全分类.第五章研究了秩为2且二阶元个数大于3的有限2-群G.我们证明了,若Ω1(G)≌D2n或D2n*C4,其中n≥3,则G,交换,且G中存在极大子群M使得M亚循环.若Ω1(G)≌D2n*Ω2m,其中n,m≥3,则Φ(G)≤Ω1(G),或｜Φ(G)｜=｜Ω1(G)｜且G'∩ Ω1(G)是Ω1(G)的极大子群.
[Abstract]:In the study of finite group theory, the normality of subgroups and some kind of commutativity between subgroups is the basic starting point for people to study, while the normalization and centralization of subgroups are a measure of the normality and Commutativity of subgroups. Thus, the study of the structure of the finite group by the regularized subgroup and the centralization of subgroups becomes very popular. In this paper, we will also study the structure of the finite p- group from the regularized subgroup and the centralization of the subgroup. At the same time, we do further research on the finite 2- group with rank 2. The third chapter studies the structure of the 2 equilibrium p- group. The finite group G is a n equilibrium group, and if D (H) is satisfied arbitrarily in G ) the subgroups of N, D (K) > n are H, K, H < NG (K) or K < NG (H). We not only completely classify the two element generation 2 equilibrium groups, but also prove that there is a regular subgroup of subgroups in the three element generation 2 equilibrium group. The fourth chapter studies the arbitrary elements of O (x) =exp (G). The fourth chapter studies the structure of the CAC-p- group. The finite p- group G is a CAC-p- group, if the non cyclic commutative subgroup H which is not included in the center of the G is CG (H) cycle. This chapter gives the complete classification of the group. The fifth chapter studies the finite group with rank 2 and the number of order of the two order more than 3 If Omega 1 (G) D2n or D2n*C4, where n is equal to 3, then G, exchange, and the existence of a maximal subgroup M in G is the subcycle of M.
【学位授予单位】：上海大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O152.1

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