Suzuki Groups And Flag-transitive Point-primitive 2-（ν, K, λ） （λ ≤10）Designs

The development of group theory has a history of more than one hundred years.We know there are important internal connections between the group theory and the theory of combinatorial designs,there are interacted on each other,which reflected by the flag-transitivity,point-primitivity and other properties of the automorphism groups.The study of the classification of designs with good transitivity is an important research area.Recently,the study of the classification of designs has many good research achievement.We assume D is a 2-(υ,κ,λ)design,in 2013,it has been proved by Tian that G is either an affine or almost simple group,if G≤Aut(D)is flag-transitive and point-primitive and A<100.Recently,Liang and Zhou researched non-symmetric designs,it is proved that for a non-symmetric 2-(υ,κ,2)design,the flag-transitive,point-primitive automorphism group of D must be an affine or almost simple group.The classification of symmetric design has been very comprehensive.However,the classification of non-symmetric design has not finished yet.This paper will continue to study flag-transitive point-primitive 2-designs,and discuss the 2-(υ,κ,λ)designs under the conditions that automorphism group is Suzuki and A<10.We obtained the result as follows:Theorem 3.0.1.If D is a non-trivial 2-(υ,κ,λ)design with A<10 which admits a flag-transitive,point-primitive automorphism group G=Sz(q).Then D is a 2-(65,8,7)design and G=Sz(8).The main result of this paper is the following:In Chapter 1,we give a survey of the background and modern development of groups and combinatorial design,and introduce the connection between them,and lift the main results of this thesis.In Chapter 2,we introduce some elementary concepts and conclusions of the group theory and combinatorial design theory,which will be used in the following chapters.In Chapter 3,we study that the classification of the flag-transitive,point-primitive automorphism group of 2-(υ,κ,λ)designs(λ≤10)and G=Sz(q),and proved theorem3.0.1.