3 - Homogeneous Group Of Flag Passed 5 - (n, K, 3) Design

At present,in the field of algebra combination,it is a hot topic in international research that:On the assumption that the automorphism groups of combinatorial designs have good transitivity(such as block-transitive and flag-transitive),we try to find out the combinatorial designs and the corresponding automorphism groups.Flag-transitivity is a quite important property in all of the properties of designs,and the object of the thesis is to research flag-transitive non-trivial5-(v,k,3) design,which relies on the classification of the finite3-homogeneous permutation groups.The thesis consists of three chapters.In chapter l,we give some introduction about the subject background,current research situation, the content we will research and methods we will use in this thesis.In chapter2,we simply introduce some basic concepts and conclusions of group theory and combinatorial design for later study.In chapter3,we first prove some important lemmas and theorems which will be used in this thesis.Then we discuss the groups of automorphisms which are affine type and almost simple type act flag-transitively on5-(v,k,3) and we get the main theorems as follows:Main Theorem1:Let D=(X,B,I)be a non-trivial5-(v,k,3)design.If G≤Aut,(D)acts flag-transitively on D,then G is not isomorphic to Affine Type,which isCase1:G=AGL(1,8),ATL(1,8),or ATL(1,32)；Case2:G0=SL(d,2),d≥2；Case3:G0=A7,v=2Main Theorem2:Let D=(X,B,I) be a non-triVial5-(v,k,3)design,and G≤Aut(D)acts flag-transitively on D.If G is Almost S imple Type,then(i) Soc(G) is not isomorphic toCase1:N=Ar,v≥5；Case2:N=PSL(2,q),q=pm,v=q+1,where p>3and p is prime；Case3:N=Mr,v=11,22,23；Case4:N=M11,y=12.(ii)if N=M12,v=12,then D is a non-trivial5一(12,6,3)design,and G-M12.(iii) if N=M24,v=24,then D is a non-trivial5一(24,6,3),5一(24,7,3)or5-(24,8,3)design,and G-M24.