Two Dimensional Classical Groups PSL(2,q)and Flag-transitive 2-(v,k,λ)Designs

From the development of algebra,we can see the finite abstract groups and the permutation groups are important in its history.With the development of combinatorial mathematics,people find an interesting thing,that is,the combinatorial design is a branch of combinatorial mathematics which is very closely related to the group theory.So an important research field-groups and combinatorial designs is born.Recent decades,people begin to do deep research on the t-(v,k,λ)design and its automorphism group.And they divide the designs into two parts,symmetric de-signs and non-symmetric designs.Nowadays,the symmetric designs are widely studied.The research on symmetric designs is more than non-symmetric designs.Nearly ten years,researchers basically completed the classification of symmetric designs under some interesting conditions.However,then they gradually turned to the research on the non-symmetric 2-(v,k,λ)designs and made some achievements.Based on these results,this thesis will study a class of the non-symmetric 2-(v,k,λ)designs.Here we assume that the automorphism group G of the design D is flag-transitive with Soc(G)(?)PSL(2,q),and we get the following result:Theorem 0.1.Let D be a nontrivial 2-(v,k,λ)design with(γ,λ)= 1 and v<2500;and let G<Aut(D)be flag-transitive with Soc(G)(?)PSL(2,q).Then,up to isomorphism,there are exactly such 12 pairs(D,G),which are listed in Table 1.Table 1:12 designs and their parameters(?)The structure of this paper is as follows:Chapter 1,it states the background and the advancement of combinatorial designs,and then explains the conclusion.Chapter 2,it describes the main results about group theory and design theory,which are the basis of the next chapter.In addition,we introduce the related lemmas which will be used in this paper.Chapter 3,it uses the related knowledge about group theory and design theory to write program.And then it analyzes the data one by one,and obtains the main result.