Block-transitive 2-designs

The classification of flag-transitive designs is a typical problem under the interaction between group and combinatorics,research on this project is already in full swing and it has become one of the forefront subjects of the finite group theory and combinatorial design theory.Block-transitivity is a much weaker condition than flag-transitivity.Even though,many results have been obtained so far.The relationship among flag-transitivity,block-transitivity,and point-primitivity is really interesting:If G is block-transitive and the block stabilizer GB acts transitively on B,then G is flag-transitive;Flag-transitive t-designs with t>3 are necessarily point-primitive;For fixed λ,there exist only finitely many 2-(v,k,λ)designs admitting a flag-transitive automorphism group which is not point-primitive;Block-transitive t-(u,λ,λ)design with v>((k2)-1)2 is necessarily point-primitive.Generally,a 2-(u,κ,λ)design with λ = 1 is called a linear space.The block-transitive linear spaces admitting an almost simple group have received much attention.In 2000,Camina and Spiezia proved that if G is an almost simple group which acts block-transitively on a linear space then Soc(G)cannot be a sporadic simple group;Camina et al.showed in 2003 that a non-trivial linear space admitting a block-transitive automorphism group G with alternating socle is necessarily PG1(3,2),and G =A7 or A8;Besides,there are also results when the socle is a classical group or simple group of Lie type.After the classification of block-transitive 2-(v,κ,λ)designs with λ = 1 and almost simple automorphism group has achieved many results,we have the ambition to com-pletely conquer the problem of the classification of block-transitive 2-(u,κ,λ)designs withλ general,although this must be a very difficult and complicated thing.This thesis is based on this aim to completely classify block-transitive 2-(u,κ,λ)designs admitting an almost simple automorphism group with sporadic socle or alternating socle.The main research work is as follows.In Chapter 1,we give a survey of the background and modern development of groups and combinatorial designs,and describe the major research content of this thesis.In Chapter 2,we introduce some elementary concepts and conclusions of the group theory and combinatorial design theory,which will be used in the following chapters.In Chapter 3,we classify 2-(v,κ,λ)(2≤λ≤10)designs which admit a block-transitive point-primitive automorphism group with sporadic socle.In Chapter 4,we show that for fixed λ,there exist only finitely many non-trivial symmetric(u,k,λ)designs with odd v which admit a point-transitive automorphism group with alternating socle.In particular,we give the classification of this type of designs when λ = 2,3,4,5,respectively.In Chapter 5,we give the necessary conditions of the existence of point-transitive 2-(81,5,1)designs and 2-(196,6,1)designs.Finally,after the summary of this dissertation,some problems are proposed for further study.