Block-Transitive Automorphism Groups Of Symmetric Designs With At Most 15 Points

There are important internal connections between the finite group theory and the theory of combinatorial designs.The study of automorphism groups of design can help us to solve the classification problems of designs or find new designs.In turn,the automorphism groups of designs can help us to understand the structure of some groups more clearly.This thesis is based on previous research,and studies the block-transitive automorphism groups of 2-(v,k,λ)symmetric designs with v ≤15.The main result is the following:Theorem 3.0.1.Let D=(P,B)be a nontrivial symmetric 2-(v,k,λ)design with v ≤15,and G ≤Aut(D)be block-transitive.Then(a)v=7,and D is a symmetric 2-(7,3,1)design,G is F21(7),or L(7).(b)v= 11,and D is a symmetric 2-(11,5,2)design,G is F55(11),or L(11).(c)v= 13,and D is a symmetric 2-(13,4,1)design,G is F39(13),or L(13).(d)v=15,and D is a symmetric 2-(15,8,4)design,G is A5(15),F(5)[1/2]S(3),GL(2,4),S5,A6,3S5,S6,A7 or PSL(4,2).The structure of this thesis is as follows:In Chapter 1,we state the backgrounds and modern developments of groups and designs,and describe the main contents of this paper.In Chapter 2,we give some basic knowledge of group theory and combinatorial design,which provide a theoretical basis for the research in the third chapter.In Chapter 3,we find out the block-transitive automorphism groups of symmetric 2-(v,k,λ)designs with at most 15 points and give the proof of Theorem 3.0.1.