Alternating Groups And Flag-transitive,Point-primitive,Non-symmetric 2-(v,k,3) Designs

Finite permutation groups form one of the important part of group theory which have great research value on some combinatorical structures,especially on the combina-torial design theory.Therefore,when we classify the designs,by using the properties of the automorphism group of designs,especially the automorphism group of the design is flag-transitive,point-primitive.Nowadays,the research of symmetric designs have been nearly mature.Some scholars gradually shifted the research work on the non-symmetric designs.In view of the above situation,thesis will focus on the non-symmetric designs with the automorphism group G is flag-transitive,point-primitive and the socle of G is an alternating group.The 2-(v,k,λ)designs with smaller A are interesting.By the O’Nan-Scot theorem,Regueiro proved that the automorphism group of the flag-transitive point-primitive of a 2-(v,k,3)symmetric design is affine or almost.Afterwards,Dong Huili and Zhou Shenglin almost completely classified such designs in a series of papers.For the non-symmetric 2-designs,Liang Hongxue and Zhou Shenglin proved that the non-symmetric 2-(υ,k,2)designs with a flag-transitive point-primitive automorphism group G and Soc(G)= An can only be a 2-(6,3,2)or 2-(10,4,2)design.In this thesis,we classified the 2-(v,k,3)designs with a flag-transitive point-primitive automorphism group G and Soc(G)= An,(n ≥ 5).The main result is the following:Theorem 3.0.1:Let D be a non-symmetric 2-(v,k,3)design.If G is flag-transitive point-primitive automorphism group of D and Soc(G)= An(n ≥ 5),then D is a 2-(5,3,3)design with G = A5 or S5.The structure of this thesis is as follows:Chapter Ⅰ,we mainly introduce the research history and current situation of group theory and combinatorial designs and the main result of this thesis;Chapter Ⅱ,we mainly do some preparatory work,introduce some basic knowledge of the group theory and combinatorial designs;Chapter Ⅲ,it is the most important part of the paper.We prove the main theorem in three steps:M acts primitively onΩn;M acts transitively and imprimitively on Ωn;M acts intransitively on Ωn,whereΩn = ｛1,2,…,n｝.