Flag-transitive Automorphism Groups Of 2-(v,k,λ) Designs

A 2-（v, k, λ） design D is an incidence structure （P, B）, where P is a set of v points, B is a set of k-element subsets of P called blocks, every block must be incident with at least two points and every pair of distinct points is incident with λ blocks exactly. An automorphism group G of D is a permutation group on P which leaves B invariant. The research about designs and groups is one of important fields of algebraic combinatorics, their interactions rely on the properties of automorphism groups. On one hand, the au-tomorphism groups which have some good properties can help us find new and classify designs. On the other hand, the study of designs in its own right give us better under-standing the structure of some groups. In this paper, we will study the classification of flag-transitive 2-（v,k,λ） designs, dealing with some special cases to find general results on the classification of designs of this type.Flag-transitivity is one of many conditions that can be imposed on the automorphism group of a design. In particular, flag-transitive symmetric designs with λ small have been studied by many scholars, and some important results have been obtained. The research of symmetrical designs has a long history. In this paper, we apply the methods which are used in symmetric designs to non-symmetric cases. Firstly, we discuss non-symmetric 2-（v,k,λ） designs, and obtain the classification of flag-transitive non-symmetric designs with （r, λ）= 1 and alternating socle. And then we study the classification of flag-transitive symmetric （v, k, λ） designs with alternating group as its socle and lager λ. Secondly, we prove that the socle of the automorphism group of a flag-transitive （v, k,4） symmetric designs can not be a simple group of Lie type. Thirdly, we discuss the classi-fication of flag-transitive point-primitive （v, k, λ） symmetric designs with general λ, and the socle of the automorphism group of D is one of 4 simple groups of Lie type. Finally, we apply the results of lager subgroups of simple groups to study flag-transitive point-primitive （v, k, λ） symmetric designs, whose automorphism group is one of simple groups of Lie type.The main results of this thesis are as follows.Theorem 3.0.1. Let D be a non-symmetric 2-（v,k,λ） design with （r, λ）= 1, where r is the number of blocks through a point. If G≤Aut（D） is flag-transitive with alternating socle, then up to isomorphism （D, G） is one of the following:（i） D is a unique 2-（15,3,1） design and G= A7 or A8;（ii）D is a unique 2-（6,3,2）design and G=A5；（iii）D is a unique 2-（10,6,5）design and G=A6 or S6.Theorem 4.0.1. If V is a 2-（v, k, A） symmetric design with （r, λ）2 < A, where r is the number of blocks through a point. If G < Aut（D） is flag-transitive with alternating socle An [n ≥ 5), then D is a 2-（15, 7, 3） design, and G = A7 or A8.Theorem 5.0.1. There is no （v, k,4） symmetric design admitting a flag-transitive, point-primitive almost simple automorphism group with exceptional socle of Lie type.Theorem 6.0.1. Let D be a symmetric 2-（v, k, A） design admitting a flag-transitive, point-primitive automorphism group G, with X（?）G ≤ Aut（X） for some nonabelian simple group X. Then X cannot be Sz（q）, ²G₂q), ²F₄（q） or ³D₄（q）.Theorem 7.0.1. Let Dbea 2-（v,k,λ） symmetric design, G < Aut（D） be flag-transitive, point-primitive. If G is isomorphic to a simple group of exceptional Lie type, then one of the following holds:...