关于1-连通特征标素因子图与相应有限群的研究

In the process of the research on the character degrees and their influence on the structure of related groups, character degree graphs (mainly contain prime graph and common divisor graph) have been the highly focused study object and tool. Based on the rich theories of both non-connected and strong connected prime graphs, in this project, we will study on 1-connected (which is connected and contains cut point)prime graph and finally classify and characterise it from graph and group aspects respectively. Specifically speaking, firstly, under known necessary conditions on prime graph, by using block graph theory and technical method on one hand, and making use of straightforward construction of related groups by using ring technical on the other hand, we classify all of the 1-connected prime graphs. Furthermore, we give more restrictions on some prime graphs with high order. Secondly, combined with the complete non-connected solvable group theory, by using the induced and extended character theory, we will character cut point from group theory, and explore the properties of finite solvable groups whose prime graphs are 1-connected. Furthermore, for finite solvable groups whose prime graphs are some 1-conneced prime graphs with terminal point, we determine their structures. The expected results of this project will enrich and further develop the character degree graph theory.

在研究有限群的特征标次数以及其对相应群结构影响的过程中，特征标次数图（主要包括素因子图与公因子图）是人们高度关注的研究对象与研究工具。基于非连通与强连通特征标素因子图方面比较丰富的研究成果，本项目拟围绕连通性介于非连通与强连通之间的1-连通(即含割点且连通)特征标素因子图进行研究，从图论与群论两方面对其分类与刻画。具体地，其一、在素因子图已有的限制条件下，借助图论中的块图理论与方法，并通过运用环技术直接构造相应群例的方式，对1-连通素因子图进行确定与分类，在此基础上对高阶的素因子图给出一些刻画条件；其二、运用特征标的诱导与扩张理论，结合非连通可解群的分类与性质，对素因子图中的割点给出群论方面的刻画，归纳探索素因子图含割点的有限可解群的性质，并进一步确定部分素因子图中含悬挂点的有限可解群的结构。本项目的预期研究成果将丰富并进一步发展特征标次数图的相关理论。

在研究有限群的特征标次数以及其对相应群结构影响的过程中，特征标素因子图是人们高度关注的研究对象与研究工具。基于非连通与强连通特征标素因子图方面丰富的研究成果，本项目对连通性介于非连通与强连通之间的1-连通(含割点且连通)特征标素因子图及相关的有限群进行了研究。具体地，一、确定了有限可解群的1-连通特征标素因子图仅有一个割点，并对割点给出一个刻画。二、证明了1-连通有限可解群（即特征标素因子图为1-连通的有限可解群）至多有两个正规非可换的Sylow子群，且给出了达到上界时群的直积结构。进一步，借助于该结果，对特征标素因子图为四边形块图的有限可解群的结构进行了研究，得到此类群也至多有两个正规非可换的Sylow子群，并确定了处于上界值时的群的直积结构。三、对直径长为3的特征标素因子图所对应的有限可解群，证明了其分别满足Gluck猜想,Isaacs-Navarro-Wolf猜想以及Takeata不等式。项目成果引起了多家外国学者的后续跟踪研究, 并进一步发展了特征标素因子图的相关理论。