Primitive Permutation Groups And Flag-transitive Non-symmetric 2-(υ,k,2) Designs

The thesis aims at the classification of the flag-transitive point-primitive non-symmetric 2-(υ,k,2)designs.In Chapter Ⅰ,we give a comprehensive survey of the research backgrounds,modern developments of groups and design theory,and main results of this thesis.In Chapter Ⅱ,we introduce some elementary definitions,and some preliminary re-sults on abstract group theory,finite permutation group and combinatorial designs which will be used in this thesis,and give a brief description of Aschbacher’s classification.In Chapter Ⅲ,based on the O’Nan-Scott theorem,we study the point-primitive automorphism groups G of flag-transitive non-symmetric 2-(υ,k,2)designs,and find out that G is an affine or almost simple group.In Chapter Ⅳ,we focus on the classification of flag-transitive point-primitive non-symmetric 2-(υ,k,2)designs with sporadic socle,and prove that there exists a unique 2-(176,8,2)design with G = HS,the Higman-Sims simple group.In Chapter Ⅴ,we move on the case that the automorphism group of a non-symmetric 2-(υ,k,2)design is flag-transitive point-primitive with socle An,n≥5,and obtain 2 designs:2-(6,3,2)design and 2-(10,4,2)design.In Chapter Ⅵ,we deal with the non-symmetric 2-(υ,k,2)designs admitting a flag-transitive and point-primitive automorphism group with socle PSL(n,q),n≥3,and get a 2-(3n-1/2,3,2)design with socle PSL(n,3).