Flag-transitive And Point-primitive 2-(v,k,λ) Non-symmetric Designs And Sporadic Simple Groups

The classification of flag-transitive designs is a typical problem between groups and combinatorial designs, and it has become one of the leading subjects of the finite group theory and combinatorial design theory. Since 1981, after finished the classification of finite simple groups, many scholars study the flag-transitive (v, k,λ) designs, and many achievements have been made. There is a deep inner relationship between the theory of combinatorial designs with some kinds of symmetry properties and the theory of finite groups, and these symmetry properties are mainly reflected by all kinds of transitivites of automorphism groups of designs. On this basis, we study the classification of the non-symmetric designs. A non-symmetric design is a design which the number of blocks are greater than points. Here we study the classification of flag-transitive non-symmetric design with λ small, and the automorphism group is point-primitive. The thesis structure is as follows:In Chapter 1, the introduction is given. In this part, the research backgrounds and current situation are stated, and describe the main results of this thesis.In Chapter 2, we introduce some basic concepts and results of the groups theory and combinatorial design theory which will be used.In Chapter 3, we study the classification of non-symmetric (v, k,5) designs. The result is as follows:Theorem 0.1. Let V=(ρ,B) be a nontrivial non-symmetric 2-(v,k,5) design which admits a flag-transitive, point-primitive automorphism group G of almost simple type and Soc(G) is a sporadic simple group. Then D and G are one of the followings:(i) D is the unique (12,22,11,6,5) design and G=M11(ii) D is the unique (22,77,21,6,5) design and G= M22 or M22:2.In Chapter 4, we continue to study non-symmetric 2-(v, k,6) designs, and the result is following:Theorem 0.2. Let D= (P,B) be a nontrivial non-symmetric 2-(v,k,6) design which admits a flag-transitive, point-primitive automorphism group G of almost simple type. Then Soc(G) is not a sporadic simple group.