Block-transitive T-designs With T Large

The thesis aims at discussing the existence, classification and construction of block-transitive t-designs for3≤t≤6.It consists of three chapters.In Chapter1, we give a comprehensive survey of the backgrounds and modern developments of groups and designs.In Chapter2, we introduce some elementary concepts, which will be used in this thesis.In Chapter3, we discuss the existence, classification and construction of block-transitive t-designs for3≤t≤6(A t-(v,k,λ) design is called a large t-design for3≤t≤6). This Chaper is consisted of four sections.Firstly, we consider the case where there is an almost simple group with socle PSL(n,q) acting on a3-(v,k,1) design. We get the following theorem.Main Theorem1Let D=(X,B) be a nontrivial simple3-(v,k,1) design and G≤Aut(D). If and q=p∫where p is a prime andfis a positive integer, then one of the following results occurs:(1). If G is block-transitive, then G is flag-transitive, and(i) D is isomorphic to a3-(pf+1,pm+1,1) design whose points are the elements of the projective line GF(pf)∪{∞} and blocks are the images of GF(pm)u{∞} under PGL(2,pf)(resp.PSL(2,pf) with f/m odd),and PSL(2,pf)≤G≤PГL(2,pf),where m|g.(ii)D is isomorphic to a3一(q+1,4,1)design whose points are the elements of the projective line GF(pf)∪{∞} with q≡兰7(mod12),and blocks are the images of {0,1,ε,∞) under PSL(2,q),where ε is a primitive sixth root of unity in GF(q),and PSL(2,q)≤G≤P∑LL(2,q),(2).If G is not block-transitive,then pf≡1(mod4),and D is isomorphic to a3一(pf+1,pm+1,1)design with2m|f,and where where B is a k-subset of GF(pf)U{∞}with P3(B)∩{0,1,∞}G=Φ,where is the set of3-subset of B.Secondly,we discuss the existence of block-transitive4-(q+1,7,λ) designs admitting PGL(2,q).We construted some4-(q+1,7,λ) designs for given parameters,and get following result.Main Theorem2Let B∈P7(X).If (X,BG)is a4一(q+1,7,λ)design,then q=16,32,17,23,37,107,and(1)If q=17,G=PGL(2,17)acts block-transitively on4一(18,7,λ) designs,then λ=28,56,and there are three non-isomorphic4一(18,7,28) designs and exactly one4一(18,7,56)design.(2)Ifq=23,G=PGL(2,23)acts block-transitively on4一(24,7,λ) designs sitively,then λ=20,40,and there are three non-isomorphic4一(24,7,20)designs and seven non-isomorphic4-(24,7,40)designs. (3)If q=37,G=PGL(2,37)acts block-transitively on4-(38,7,λ) designs,then λ=24,and there are three non-isomorphic4-(38,7,24) designs.Thirdly,we consider the flag-transitive5-(v,k,2)designs,get following result.Main Theorem3Let D=(X,B)be a non-trivial5-(v,k,2)design.Then PSL(2,2n)≤Aut,(D) cannot acts flag-transitively on D.Fourthly,we began consider the block-transitive6-designs.In this section,we discuss flag-transitve6-designs,and get the following result.Main Theorem4Let D=(X,B)is a nontrivial simple6一(v,k,λ) design and G≤Aut(D).If G is flag-transitve,then λ≥5.Finally,We consider a conjecture of Cameron and Praeger that there are no nontrivial block-transitive6-designs.We proved that there are no nontrivial block-transitive6-designs for k≤10,and get the following result.Main Theorem5Let D=(X,B)be a non-trivial6-(v,k,λ)design with k≤10000,and PSL(2,q)≤G≤PГL(2,q),v=q+1,q=pc,p is prime.Then G cannot acts block-transitively on D.