2-(v,k,1) Design And Two-dimensional Projective Linear Group

Abstract:Let D be a2-(v,k,l) design, G=PGL(2,q) q>3, q=pf(p is a prime) is block-transitive automorphism group. We discuss when G acts transitively,the design whether there is, if there is the ability to find the specific design or draw the design of the form.In this paper, from the following four aspects:First, we begin by reviewing some of the basic concepts of group theory and combinatorial design theory, the contact group theory and combinatorial design theory, group theory and combinatorial design theory research status quo and have come to some conclusions.Second, we have come to some conclusions on the basis of when the2-dimensional protective linear group G=PGL(2,q),q>3, q=pf,(pis a. prime) district passed role in the2-(v,k,l) design, we prove an important theorem.Let G is the automorphism group,2) is a2-(v, k,l) designed and D non-projective plane, G=PGL(2,q),q=pf>3, if G is the block-transitive, then it is point primitive, Ga is isomorphic to NG(P), Zq-1:Z2or Zq-1:Z2,here P is a Sylow-p subgroup of G. Third, we discuss the2-dimensional projective linear group G=PGL(2,q),q>3, q=pf,(pis a prime) block-transitive role in the2-(v,k,l) design, the existence and form of the2-(v,kl,l) design. Concrete proof we based the form of Ga and the number of conjugacy classes in GL from two aspects:(a) If GL has only a conjugate of class, we prove that the2-(v,k,l) design does not exist at this time; (b)If GL has at least two pairs together conjugacy classes.①GL,≌S4;②GL≌D4;③GL≌D2d(d>2,2d|(q±1));④GL≌D2d(d>2,d|(q±1),2d|,(q±1));⑤GL≌GL(2,pm),m|f;Among,Gα={g∈G|αg=α},α∈P, GL={g∈G|Lg=L},L∈B.is point stabilizer subgroup and district stabilizer subgroup of G.After a discussion of the research,we have:Only when GL≌S4,Gα≌Z(q-ε):Z2(ε=±1), We have a2一(136,10,1)design. Fourth to complete the proof of the main theorem, and related knowledge development and promotion of research work to make a simple point.Mainly include the following three aspects:First,our study is when G一PGL(2,q)(q=pf, p is a prime numbers q is odd and q>3) block-transitive role in the2-(V,k,l) design,the existence of2-(v,k,l) design and the form of existence.This problem we can further enlarge as when G=PΓL(2,g)(Soc(G)=PGL(2,g)) block-transitive role in the2-(v,k,l) design,the existence of the2-(v,k,l)design and form of existence.Second,our research work limit is a2-(v,k,l)design,that be limited to some of the parameters of the design.We can put some of the wide side defined parameters.For example,2-(v,k,λ) design,or simply a case of different λ given value.At last,above①,after calculations prove that there is a2-(136,10,1) design,we just come to this design to meet the given number of relations, but we do not have to verify the existence of the design.