Simple 3-designs And Two-dimensional Projective Special Linear Group

The study of the existence and construction of simple t-designs is an important problem in the field of combinatorial designs.For t=2,the results of simple 2-designs are already very rich.However,for t≥3,the results of simple t-designs are much less than simple 2-designs.For example,when t>4,the results of simple t-designs are sporadic.Moreover,the commonly used methods of the construction of simple 2-designs can’t apply to the construction of simple t(t≥3)-designs.In the results of many simple t(t≥3)-designs,scholars mainly use algebraic methods,especially using orbits of the multiply transitive group to construct simple t-designs.In this thesis,since the projective special linear group PSL(2,q)is a multiply transitive group,and we are very familiar with its subgroup structure,we mainly use the orbits of PSL(2,q)to construct simple 3-designs.This thesis is divided into four chapters.The first chapter is the summarization of the dissertation.We discuss the background and current situation of this subject,and give the main results of this thesis.In the second chapter,we mainly introduce the elementary concepts of the group theory and t-designs,that will be used in this thesis.Then we construct the basic theory system of this thesis.The third chapter shows all simple 3-designs with block size 9 with PSL(2,2n)as an automorphism group.In the fourth chapter,we introduce all simple 3-designs with block sizes d+1,d and d+2,with PSL(2,q)as an automorphism group.