Classification Of Flag-transitive Point-primitive Symmetric Designs

The classification of flag-transitive designs is a typical problem between groups andcombinatorial designs, and it has become one of the leading subjects of the finite grouptheory and combinatorial design theory. As a part of this project, the classification offlag-transitive symmetric (v, k, λ) designs is underway. In1987, Kantor classified flag-transitive symmetric (v, k,1) designs which are called projective planes. After that, byusing the O’Nan-Scott theorem and the classification theorem of finite simple groups,Regueiro and Zhou et al. reduced the classification of flag-transitive symmetric (v, k,2)and (v, k,3) designs to the situation that the automorphism group is a1-dimensionalafne group.After finished the classification of flag-transitive symmetric (v, k, λ) designs with λsmall, we have the ambition to solve the problem on the classification of flag-transitivesymmetric (v, k, λ) designs with general λ, although this is a very difcult and compli-cated problem. This thesis is focus on this aim, and completely classified flag-transitivesymmetric (v, k, λ) designs admitting a point-primitive automorphism group G of almostsimple type, with a sporadic group or P SL(2, q) as its socle. The structure of this thesisis as follows.In Chapter1, we give a survey of the backgrounds and modern developments ofgroup and design theory, and describe the main research content of this thesis.In Chapter2, we introduce some elementary concepts and results of the group theoryand combinatorial design theory which will be used in this thesis.In Chapter3, by using the O’Nan-Scott theorem, we get the following theorem, andthen propose a conjecture for general λ.Theorem3.0.1. Let D be a symmetric (v, k, λ) design with λ≤100which admitsa flag-transitive, point-primitive automorphism group G. Then G is of almost simple orafne type.Conjecture3.0.1. If a symmetric (v, k, λ) design D admits a flag-transitive point-primitive automorphism group G, then G must be an afne or almost simple group.In Chapter4, we study the classification of symmetric (v, k, λ) designs admitting aflag-transitive, point-primitive automorphism group G of almost simple type with spo-radic socle. The result is that there are only6symmetric designs up to isomorphism. Theorem4.0.1. Let D=(P, B) be a symmetric (v, k, λ) design with a flag-transitive, point-primitive automorphism group G of almost simple type and Soc(G) isa sporadic simple group. Then the design has parameters (144,66,30),(176,50,14),(176,126,90), or (14080,12636,11340).In Chapter5, we completely classify symmetric (v, k, λ) designs admitting a flag-transitive, point-primitive automorphism group G of almost simple type with two dimen-sional classical group P SL(2, q) as its socle. The main result is as follows:Theorem5.0.1. Let D=(P, B) be a symmetric (v, k, λ) design which admitsa flag-transitive, point-primitive automorphism group G. If G is of almost simple, thatis, X G≤Aut(X) where X=Soc(G)～=P SL(2, q), then the design has parameters(7,3,1),(7,4,2),(11,5,2),(11,6,3), or (15,8,4).In Chapter6, we give a brief introduction about the classification of flag-transitivesymmetric (v, k, λ) designs admitting a point-primitive automorphism group G of almostsimple type with P SL(12,2) or Anas its socle. Finally, some problems are proposed forfurther study.