Classification Of Block Transitive 2-(v,8,1) Designs

This thesis is devoted to the classification of block transitive 2-(v,8,1)designs.It consists of four chapters.In the first chapter,the research background and development status of finite groups and block designs are comprehensively summarized.In the second chapter,the basic knowledge of finite groups and block designs are introduced.The classification of block transitive 2-(v,3,1)designs was completed about 45 years ago.In 1996,Camina and Siemons classified 2-(v,4,1)designs with a block transitive,solvable groups of automorphisms.Li classified 2-(v,4,1)designs admitting a block transitive,unsolvable automorphism groups.Li and Tong classified 2-(v,5,1)designs with a block transitive,solvable groups of automorphisms.Han and Li classified 2-(v,5,1)designs with a block transitive,unsolvable groups of automorphisms.Camina confirmed:if the automorphism groups of 2-(v,k,1)are block transitive and point primitive,then the socle of G are either elementary commutative groups or noncommutative simple groups.This result provides a research idea:the classification theorem of finite single groups can be used for studying the classification of 2-(v,k,1)designs.Through the exploration of experts at home and abroad,plentiful results are obtained from block designs,especially from 2-(v,k,1)designs.In the third chapter,we study the automorphism groups of 2-(v,8,1)designs,the theorems are as follows:Theorem 3.1 Let D be a 2-(v,8,1)design with G≤Aut(D)block transitive,point primitive but not flag transitive,then the socle of G is not the classical simple group in the finite field GF(q).By theorem 3.1,the classification of block transitive 2-(v,8,1)designs was completed.And we haveTheorem 3.2 Let D be a 2-(v,8,1)design with G≤Aut(D)block transitive but not flag transitive,then G is solvable group and G≤AΓL(Y,v)(v-pn).In the fourth chapter,we summarize the main results of this thesis,and put forward some new problems for the future study of t-(v,k,λ) designs.