The Classification Of Flag-transitive 2-designs With Prime Parameters

This paper studies the classifications of the flag-transitive 2-designs.This problem originates from the well-known result of classification on the flag-transitive linear space(the 2-(v,k,1)designs),which is done by six mathematicians(F.Buekenhout,etc.).Since then,many scholars turn to study the classification problem on other conditions.So far,many parts of the classification on flag-transitive 2-(v,k,2)designs are done.Until 2020,the classification of the flag-transitive 2-designs with(r,?)=1 are almost finished.We study further the classification problem on flag-transitive 2-designs with some prime parameters.In particular,we study the classification problem on condition that r is the square of a prime number or the third power of a prime number.Moreover,we study the condition that k is a prime number 5.The dissertation contains five parts:Chapter ? is the introduction.We first introduce the historical origin of the classification problems on 2-designs.Then we present some major research on the classification of flag-transitive 2-designs in the last several decades.Later we introduce our work.Chapter ? consists of four parts.In the first part,we show the main concepts and properties of combinatorial designs.In the second part,we show the basic theory of finite permutation groups.In the next part,we introduce the well-known O'Nan-Scott Theorem and the classification theorem of primitive groups of odd degree.In the fourth part,we focus on the basic theory of automorphism groups of a 2-design.In Chapter ?,we present our work on flag-transitive 2-designs D where r is the square or the third power of a prime number.If G is a flag-transitive automorphism group of a symmetric design D with prime square replication number,then G is pointprimitive of affine or almost simple type.If r is the third power of a prime number and G is point-primitive,then G is of affine,almost simple,or product type whose socle is the direct product of two non-abelian simple groups.In Chapter IV,we continue the classification work on the situation r is a prime square.We ask G to be an almost group with an alternating socle An,where n? 5.We show that D is the unique 2-(10,6,5)design and G=A6 or S6.In the last Chapter,we study flag-transitive 2-(v,5,?)designs.We reduce G to primitive group of affine or almost simple type.Moreover,we classify the case that G has a socle of sporadic simple group.We finally obtain 20 2-(v,5,?)designs and the socle of G is one of the following:M11,M12,M22,M23,M24,HS or Co3.