素数幂阶子群的嵌入与有限群的结构

发布时间：2018-05-26 17:43

  本文选题：弱M-可补性 + S-半置换性　； 参考：《上海大学》2017年博士论文

【摘要】：在有限群论中,应用子群在有限群中的嵌入性质来研究有限群的结构是人们非常感兴趣的课题.由于子群的正规性与可补性是有限群论中最基本的性质,于是由此引出许多与它们相关的嵌入性质,并且人们已经获得大量的研究成果.本文将继续这一领域的研究,重点研究素数幂阶子群在有限群中的嵌入性质,获得了一些有价值的成果.在第三章,我们主要利用素数幂阶子群的弱M-可补性,获得了有限群G的某个正规子群包含在G的F-超中心ZF(G)中的一些充分条件,其中F是一个包含超可解群类u的可解饱和群系.在第四章,我们通过研究Op(G)的d阶非循环子群的S-半置换性,获得了包含在有限群G的某个正规P-子群中的C-主因子都循环的充分条件及G的结构的相关信息,其中d为素数P的某个方幂.在第五章,我们首先引入子群的一个新的嵌入性质,即:Π*-性质,然后从素数幂阶子群的Π*-性质出发,获得了有限群G的某个正规子群E在G中p-超循环嵌入的充要条件,其中P ∈ π(E).作为上述研究的继续,我们考察了有限群G的Ap-1-剩余GAp-1,其中Ap-1是幂指数整除P- 1的交换群类.我们不仅给出了GAp- 的基本性质,而且通过研究Op(GAp-1)的给定阶特殊子群在G中的嵌入性质,获得了有限群G为P-超可解群等若干结果.
[Abstract]:In finite group theory, it is of great interest to study the structure of finite groups by applying the embedding property of subgroups in finite groups. Since the normality and complementarity of subgroups are the most basic properties in finite group theory, many embedded properties related to them are derived, and a lot of research results have been obtained. In this paper, we will continue to study the embedding properties of prime power order subgroups in finite groups, and obtain some valuable results. In chapter 3, we obtain some sufficient conditions for a normal subgroup of a finite group G to be contained in the F-supercenter of G by using the weak M-complement of a prime power subgroup. Where F is a solvable saturated formation containing the class u of a supersolvable group. In chapter 4, by studying the S- semipermutation of non-cyclic subgroups of order d of Opg, we obtain the sufficient conditions of C- principal factor cycles in a normal P- subgroup of a finite group G and the relevant information about the structure of G. Where d is a power of the prime number P. In chapter 5, we first introduce a new embedding property of a subgroup, that is, a 蟺 -property. Then we obtain a sufficient and necessary condition for a normal subgroup E of a finite group G to be phypercyclic embedded in G. Where P 鈭，

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