The Classification Of Flag-transitive 2-(v,k,?) Designs

The finite permutation groups have great applications in the study of certain com-binatorial structures,especially in the field of block designs,codes,association schemes and graph theory.The construction of 2-(v,k,?)designs with highly symmetric prop-erties,f'or example,block or flag-transitivity,is one of the leading subjects with great theoretic significance and application background in design theory.Most of the studies of flag-transitive 2-(v,k,?)designs hitherto have concerned the cases finite linear spaces and symmetric designs with A small.Thus,for the case non-symmetric designs with general A becomes an interesting problem.Base on these ideals,we study flag-transitive non-symmetric 2-(v,k,?)designs with??(r,?)2,triple systems,2-(v,k,?)designs in this thesis.Chapter I contains some research background,modern developments of group and design theory,and main results of this thesis.Chapter II collects some notations and definitions,and quotes some preliminary results on abstract group theory,permutation group and combinatorial designs which will be used in this thesis.Chapter III studies the flag-transitive automorphism groups of 2-(v,k,?)designs with ??(r,?)2and an application to symmetric designs.Chapter IV gives a classification of flag-transitive non-symmetric 2-(v,k,?)designs with??(r,?)2and sporadic socle.We proved that there are exactly such 33 non-isomorphic designs.Chapter V deals with the flag-transitive triple systems.We prove that if G is a flag-transitive automorphism group of a triple system,then G is point-primitive,and hence it must be of affine or almost simple type.Furthermore,we almost determine all triple systems which admit a flag-transitive automorphism group of almost simple type.Chapter VI discusses the 2-(v,k,?)designs admitting a flag-transitive and point-imprimitive automorphism group,and determine all possible 2-designs of this type.