Flag-transitive Non-symmetric 2-designs With At Most 50 Points

There are a long history for the study of group theory.The classification of designs mostly can be solved by means of studing their automorphism groups,because of the close relationship of groups and combinatorial designs.The classification of flag-transitive designs is a typical problem on groups and designs.Recently,the study of symmetric 2-designs has reached its peak,non-symmetric 2-designs also have many results.This paper will continue to study the classification of flag-transitive non-symmetric designs,and discuss the 2-(υ,k,λ)designs under the conditions that(r,λ)= 1 and υ ≤ 50,where r is the number of blocks through a given point.A non-symmetric design is a block design such that the numbers of points less than blocks.According to the group theory structure and the quantitative relations of the parameters of the designs,we study non-symmetric 2-(υ,k,λ)designs admitting the flag-transitive,point-primitive automorphism group.The main result of this paper is the following:Theorem 3.0.1:Let D =(P,B)be a nontrivial non-symmetric 2-(υ,k,λ)design with υ ≤ 50,and G ≤ Aut(D)be flag-transitive.If(r,λ)= 1,then there are exactly 79 non-isomorphic designs,and the pair(D,G)can only be one of 216 cases listed in Table 3-1.The structure of this thesis is as follows:In Chapter 1,we state the backgrounds and modern developments of groups and designs,and describe some results related to this thesis.In Chapter 2,we give some basic knowledge of group theory and combinatorial design which is needed in the proof of the main results.In Chapter 3,by the O’Nan-Scott theorem,we classify all flag-transitive non-symmetric 2-(υ,k,λ)designs with less than 50 points and(r,λ)= 1,and give the proof of Theorem 3.0.1.