子群的广义正规嵌入性和广义正交图的自同构群
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摘要
群论是抽象代数学中的一个重要分支,利用子群的性质来研究和刻画整个群的性质与结构一直是群论研究的一个重要课题.同时,研究代数结构,我们往往希望它能够跟图论相结合,无论是用代数的方法去研究图还是用图论的方法去研究代数结构,都是十分有意义的.在该领域,研究图的自同构群一直是一个非常重要且十分活跃的课题.
本学位论文对以上两方面进行研究,内容大致可分为两部分：一)子群的广义正规嵌入性对有限群结构的影响；二)广义正交图的自同构群.
在第3章,我们引入了几乎SS-嵌入子群的新概念,它是正规子群,S-拟正规子群,S-拟正规嵌入子群,c-正规子群以及s-嵌入子群等概念的推广.我们研究了子群的几乎SS-嵌入性与有限群结构的关系,给出了有限群为p-幂零群和p-超可解群的新的特征性定理,由此推广了一些已有结论.第4章,基于Φ-可补子群以及SΦ-可补子群的概念,我们给出了n-Φ-嵌入子群的概念及其基本性质,讨论了特定极大子群在满足n-Φ-嵌入性条件下有限群的结构,这也说明该思想方法为研究有限群提供了新的有效工具.第5章,我们在前人研究的基础上进一步研究(?)C-子群对有限群结构的影响,分别讨论了同阶子群以及某些极小子群在满足一些给定条件下的有限群的结构,得到一个群属于某些群类和一个群为幂零群的一些判别准则.
第6章,在正交图的基础上,我们利用正交空间中的m-维全迷向子空间或m-维全奇异子空间作为顶点集并恰当定义邻接关系,分别构作了奇特征和特征为2的广义正交图,本文中分别把它们简记为Γ和Γ'.6.1节主要研究特征为奇数的广义正交图的自同构群,我们首先给出了图中任意两点间的距离公式并讨论了Γ1(M)和Γ2(M)中顶点的形式及性质,其中M是一个给定顶点,Γk(M)表示顶点集{x∈V(Γ)|d(M,X)=k}此外本节还给出该图中的两类局部结构：极大集与拟四面体结构.在6.2节,我们讨论了特征为2的广义正交图的自同构群.类似于6.1节,我们也研究了Γ'1(M)和Γ'2(M)的性质,并讨论了当k≥2时次成分Γ'k(M)之间顶点的邻接关系,其中M是Γ'中一给定顶点.利用有限群,有限域,矩阵几何等工具我们确定了广义正交图Γ和Γ'的自同构群.
The theory of groups is an important branch of abstract algebra. It has been one of important topics to study the structure of finite groups by using the properties of their subgroups. At the same time, when studying an algebraic structure we often hope that it can be combined with graph theory. It is of great significance either we use algebraic method to study graph or use graph theory to study algebraic structure. In this area, it has always been a very significant and active topic to investigate the automorphism group of a graph.
We mainly study two aspects above-mentioned in this dissertation, which is di-vided into two parts:1) the influence of the generalized normal embeddability of sub-groups on the structure of finite groups;2) the automorphism groups of the generalized orthogonal graphs.
In Chapter3, we introduce the new concept of nearly SS-embedded subgroup, which is a generalization of the concepts of normal subgroup, S-quasinormal sub-group,S-quasinormally embedded subgroup, c-normal subgroup and s-embedded sub-group and so on. We investigate the relationship between nearly SS-embedded sub-group and the structure of finite groups and give some new theorems about p-nilpotency and p-supersolvability of finite groups, from which we generalize some known result-s. In Chapter4, on the base of the concepts of Φ-supplemented subgroup and SΦ-supplemented subgroup, we give the concept of n-Φ-embedded subgroup and their elementary properties, and discuss the structure of finite groups on the assumption that some maximal subgroups are n-Φ-embedded subgroup. It has been also proved that this idea provides a new effective tool for the research of finite group. In Chapter5, on the basis of previous research, we further investigate (?)C-subgroup and the structure of finite groups. We discuss the structure of finite groups on the assumption that the same order subgroups and some minimal subgroups satisfy given conditions and obtain some new criterions that a group belongs to some classes of finite groups and a group is nilpotent group.
In Chapter6, on the base of orthogonal graphs we construct the generalized or-thogonal graphs of odd characteristic and characteristic2, respectively by taking all the m-dimensional totally isotropic subspaces and m-dimensional totally singular sub-spaces of the orthogonal space, we simply denote them by Γ and Γ'. In section6.1, we study the automorphism group of the generalized orthogonal graph of odd charac-teristic. We firstly give the distance formula between any two vertices and discuss the forms of vertices in Γ1(M),Γ2(M) and the property of them, where M is a given ver-tex and Γk(M) represents the vertex set {X∈V(Γ)|d(M, X)=k}. Besides, in this section we give two types of local structure, that is, maximal set and quasi-tetrahedron. In section6.2, we investigate the automorphism group of the generalized orthogonal graph of characteristic2. Similar to section6.1, we also study the properties of Γ'1(M) and Γ'2(M) and discuss the adjacency relationship of the subconstituent Γ'k(M) when k≥2, where M is a given vertex in Γ'. By using the tool of finite group, finite field and matrix geometry we determine the automorphism groups of the generalized orthogonal graphs Γ and Γ'.
本学位论文对以上两方面进行研究,内容大致可分为两部分：一)子群的广义正规嵌入性对有限群结构的影响；二)广义正交图的自同构群.
在第3章,我们引入了几乎SS-嵌入子群的新概念,它是正规子群,S-拟正规子群,S-拟正规嵌入子群,c-正规子群以及s-嵌入子群等概念的推广.我们研究了子群的几乎SS-嵌入性与有限群结构的关系,给出了有限群为p-幂零群和p-超可解群的新的特征性定理,由此推广了一些已有结论.第4章,基于Φ-可补子群以及SΦ-可补子群的概念,我们给出了n-Φ-嵌入子群的概念及其基本性质,讨论了特定极大子群在满足n-Φ-嵌入性条件下有限群的结构,这也说明该思想方法为研究有限群提供了新的有效工具.第5章,我们在前人研究的基础上进一步研究(?)C-子群对有限群结构的影响,分别讨论了同阶子群以及某些极小子群在满足一些给定条件下的有限群的结构,得到一个群属于某些群类和一个群为幂零群的一些判别准则.
第6章,在正交图的基础上,我们利用正交空间中的m-维全迷向子空间或m-维全奇异子空间作为顶点集并恰当定义邻接关系,分别构作了奇特征和特征为2的广义正交图,本文中分别把它们简记为Γ和Γ'.6.1节主要研究特征为奇数的广义正交图的自同构群,我们首先给出了图中任意两点间的距离公式并讨论了Γ1(M)和Γ2(M)中顶点的形式及性质,其中M是一个给定顶点,Γk(M)表示顶点集{x∈V(Γ)|d(M,X)=k}此外本节还给出该图中的两类局部结构：极大集与拟四面体结构.在6.2节,我们讨论了特征为2的广义正交图的自同构群.类似于6.1节,我们也研究了Γ'1(M)和Γ'2(M)的性质,并讨论了当k≥2时次成分Γ'k(M)之间顶点的邻接关系,其中M是Γ'中一给定顶点.利用有限群,有限域,矩阵几何等工具我们确定了广义正交图Γ和Γ'的自同构群.
The theory of groups is an important branch of abstract algebra. It has been one of important topics to study the structure of finite groups by using the properties of their subgroups. At the same time, when studying an algebraic structure we often hope that it can be combined with graph theory. It is of great significance either we use algebraic method to study graph or use graph theory to study algebraic structure. In this area, it has always been a very significant and active topic to investigate the automorphism group of a graph.
We mainly study two aspects above-mentioned in this dissertation, which is di-vided into two parts:1) the influence of the generalized normal embeddability of sub-groups on the structure of finite groups;2) the automorphism groups of the generalized orthogonal graphs.
In Chapter3, we introduce the new concept of nearly SS-embedded subgroup, which is a generalization of the concepts of normal subgroup, S-quasinormal sub-group,S-quasinormally embedded subgroup, c-normal subgroup and s-embedded sub-group and so on. We investigate the relationship between nearly SS-embedded sub-group and the structure of finite groups and give some new theorems about p-nilpotency and p-supersolvability of finite groups, from which we generalize some known result-s. In Chapter4, on the base of the concepts of Φ-supplemented subgroup and SΦ-supplemented subgroup, we give the concept of n-Φ-embedded subgroup and their elementary properties, and discuss the structure of finite groups on the assumption that some maximal subgroups are n-Φ-embedded subgroup. It has been also proved that this idea provides a new effective tool for the research of finite group. In Chapter5, on the basis of previous research, we further investigate (?)C-subgroup and the structure of finite groups. We discuss the structure of finite groups on the assumption that the same order subgroups and some minimal subgroups satisfy given conditions and obtain some new criterions that a group belongs to some classes of finite groups and a group is nilpotent group.
In Chapter6, on the base of orthogonal graphs we construct the generalized or-thogonal graphs of odd characteristic and characteristic2, respectively by taking all the m-dimensional totally isotropic subspaces and m-dimensional totally singular sub-spaces of the orthogonal space, we simply denote them by Γ and Γ'. In section6.1, we study the automorphism group of the generalized orthogonal graph of odd charac-teristic. We firstly give the distance formula between any two vertices and discuss the forms of vertices in Γ1(M),Γ2(M) and the property of them, where M is a given ver-tex and Γk(M) represents the vertex set {X∈V(Γ)|d(M, X)=k}. Besides, in this section we give two types of local structure, that is, maximal set and quasi-tetrahedron. In section6.2, we investigate the automorphism group of the generalized orthogonal graph of characteristic2. Similar to section6.1, we also study the properties of Γ'1(M) and Γ'2(M) and discuss the adjacency relationship of the subconstituent Γ'k(M) when k≥2, where M is a given vertex in Γ'. By using the tool of finite group, finite field and matrix geometry we determine the automorphism groups of the generalized orthogonal graphs Γ and Γ'.
引文
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