The Automorphism Groups Of 2-(v,k,1)Designs With Good Transitivity

Since the problem of classification of finite simple groups has been solved in the early 1980s,the study of finite groups has undergone great changes.Scholars have begun to pay attention to the relationship between group and combinatorial structure,such as group and graph,group and block design,group and lattice.On the research of group and block design,so far,scholars have basically solved the classification of flag-transitive 2-(v,k,l)design.And through the exploration of experts at home and abroad,the classification of block-transitive 2-(v,k,1)design has also been a great degree of solution.The research of this thesis consists of two parts:firstly,the study of block-transitive automorphism groups of 2-(v,15,1)designs of which the socle is classical simple group,and the second is the 2-(v,k,1)design whose automorphism group is classical simple group PSpn(q).The main contents are as follows:The first and second chapters introduce the research background and current situation of finite group theory and block design,as well as some basic concepts and properties.In the third chapter,we study the automorphism groups of 2-(v,15,1)designs of which the socle is classical simple group,and get the following conclusion:Let(?)be a 2-(v,15,1)design,G?Aut((?))be block-transitive,point-primitive but not flag-transitive,then Soc(G),the socle of G,is not classical simple group over finite field GF(q).And thus get the complete classification of block-transitive 2-(v,15,1)designs.In the fourth chapter,we study the 2-(v,k,1)designof the automorphism group as classical simple group PSpn(q),and get the following conclusion:Let(?)=((?),(?))be a 2-(v,k,l)design,G?Aut((?))be block-transitive,point-primitive but not flag-transitive.If G = PSpn(q),with q even and n?14,then one of the following holds:(1)Gp?(?)and Gp is not type Spm(q)?Spn-m(q),with m?4;(2)Gp ?(?).In the fifth chapter,we summarize the main contents of this paper,and put forward some new problems for the future more in-depth study of t-(v,k,?)designs.