Sporadic Groups And Flag-transitive,point-primitive, Non-symmetric 2-(v，k，4) Designs

Group theory and combinatorial design have a strong bond, hence a certain design is often determined by the property of its automorphism group. Currently, the study of symmetric 2-designs has reached its peak, non-symmetirc 2-designs gradually becomes the focus. The aim of this paper is to classify the flag-transitive, point-primitive non-trivial non-symmetric 2-designs, whose automorphism group is almost simple with with sporadic socle.In 1987, Davies proved that if the socle of G is a sporadic simple group then G does not act flag-transitively on a 2-(v, k,1) design. Later, A six-person team made by Buekenhout, Delandtsheer, Doyen, Kleidman, Liebeck, Saxl classified flag-transitive finite linear spaces. In 2015, Tian and Zhou completed the classification of symmetric (v, k, λ) designs admitting a flag-transitive, point-primitive automorphism group G of almost simple type with sporadic socle. It is natural to study the classification of non-symmetirc 2-design with sporadic socle instead.In this paper, we use the flag-transitive and point-primitive conditions as long as parameters equations of non-symmetric 2-designs to study flag-transitive, point-primitive 2-(v, k,4) designs. The main result of this paper is the following:Theorem:If D is a non-symmetric 2-(v,k,4) design and G is a flag-transitive point-primitive automorphism group of D, then Soc(G) is not a sporadic simple group.The structure of this thesis is as follows:The first chapter is an introduction. We state the background and the research results related to this thesis.In the second chapter, we give some basic knowledge which is needed in the proof of the main results.In Chapter Three, we prove the main theorem in three steps. First of all we give the algorithm of searching potential non-symmetric 2-(v, k,4) designs. Secondly, we rule out 36 possible automorphism groups by the methods of group and design theory. Finally, we eliminate the last 2 sophisticated cases step by step with the aid of computer. Then we finish the proof of the main theorem.