The Classification Of Flag-transitive Point-quasiprimitive Automorphism Groups Of 2-Designs

Over the past twenty years,the characterization of geometric or combinatorial struc-tures in terms of their automorphism groups has attracted numerous scholars' concern and obtained many important results,especially in the fields of graph theory,design the-ory,coding theory,cryptography and so on.In this thesis,we study the classification of flag-transitive point-quasiprimitive 2-designs,namely,2-(u,k,?)designs which admit an automorphism group which is transitive on the set of incident point-block pairs and is quasiprimitive on the set of points.The starting point in our research is an example which first appeared in the work of Praeger and Zhou,they show that there exists an unique 2-(15,8,4)design which admitting a flag-transitive point-imprimitive automorphism group S5.A permutation group G on a finite set ? is said to be quasiprimitive on ? if each of its nontrivial normal subgroups acts transitively on ?.It would be easy to see that primitivity implies quasiprimitivity and the concept of quasiprimitivity is weaker than that of primitivity.The O'Nan-Scott Theorem shows that each quasiprimitive group is permutationally equivalent to one of the following eight types:(?)Holomorph affine type;(?)Holomorph simple type;(?)Holomorph compound type;(?)Almost simple type;(?)Twisted wreath product type;(?)Simple diagonal type;(?)Compound diagonal type;(?)Product action type.The quasiprimitive types in most cases are similarly to the primitive types from the O'Nan-Scott Theorem.Indeed,types(?)-(?)are certainly primitive.The O'Nan-Scott Theorem for quasiprimitive groups is very useful for our study since we have to consider the structures of the flag-transitive point-quasiprimitive auto-morphism groups of 2-designs,that is to say,the reduction theorem.This thesis consists of four chapters.In Chapter ?,we give a survey of research backgrounds,modern developments of permutation groups and combinatorial designs,and main results of this thesis.In Chapter ?,based on the O'Nan-Scott Theorem for quasiprimitive groups,we study the structures of the automorphism groups G of flag-transitive point-quasiprimitive 2-(u,k,?)designs with A<4 and prove that G is of holomorph affine or almost simple type.Using this fact,it is natural to apply the classification of finite simple groups studying of 2-(u,k,?)designs with A<4 admitting a flag-transitive quasiprimitive,but imprimitive automorphism group.And prove that there are exactly such 2 non-isomorphic designs.In Chapter ?,we study the point-quasiprimitive automorphism groups G of flag-transitive 2-(u,k,5)designs,and also find out that G is a holomorph affine or almost simple group.In Chapter IV,we deal with the 2-(u,k,?)designs with ??(r,?)2 admitting a flag-transitive automorphism group of product action type,here u=?2,and prove that G?H(?)S2,where H is a 2-transitive group,and Soc(G)?A?×A?.