The Generalized Normal Embeddability Of Subgroups And The Automorphism Group Of Generalized Orthogonal Graph

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’.