Block-designs With Some Particular Transitivity

The thesis aims at discussing the classification of block-designs, which have some particular transitivity, and consists of six Chapter.In Chapter 1, we give a comprehensive survey of the backgrounds and modern developments of groups and designs.In Chapter 2, we introduce some elementary concepts, which will be used in this thesis.In Chapter 3, basing on Huber's works, we further consider the classification of more general flag-transitive designs, and getMain Theorem 1 Let D= (X, B, I) be a non-trivial 5-(v, k,2) design. Then if G is a flag-transitive automorphism group of a designs D, then PSL(2,q)≤G≤Aut(PSL(2,q)), where q= pe and p= 2 or 3.In Chapter 4, basing on E.O'R.Regueiro's works, we further consider the classifica-tion of triplanes, and get following resultMain Theorem 2 If a (v,k,3)-symmetric design D admits a primitive, flag-transitive automorphism group G of almost simple type with classical socle T, then D is either the unique (11,6,3)-symmetric design or the unique (45,123)-symmetric design, and G (?) PSL(2,11) or G (?) PSp4(3):2 (?) PSU4(2):2, respectively.In Chapter 5, we main consider the classification of finite linear-space admitting a line-transitive automorphism group, and T≤G≤Aut(T), with T is a non-abelian simple group. Should point that when the Lie rank is small, we often reduce G block-transitive to socle T. This paper is an exploration of the reduction. We prove the following theorem:Main Theorem 3Let G act line-transitively on a linear space S= (P,(?)),and (?)G(?) Aut(L(q)) with L(q) is a Lie simple group on finite field GF(q). If T(?) F4(q), and T is not line-transitive, then TL is not the subgroups of2F4(q), B4(q), D4(q).S3,3D4(q).3, F4((?)) or the parabolic subgroups of T, where T= L(q).We are all know, in [28] Liu Weijun classified the 2-(v,7,1) with solvable block-transitive automorphism group. Therefore, it is necessery to classify the same designs with unsolvable block-transitive automorphism group. In Chapter 6, we discuss them, and get following result.Main Theorem 4 If G is a group of automorphism of a 2-(v,7,1) design D,and is block-transitive, unsolvable and imprimitive, then G≠PSL(n,q), where q is odd and (n,q)≠(2,2),(2,3).The projective special linear group PSL(2, q) is often use for constructing t-designs. In Chapert 7, we investigate the existence of simple 3-designs with block size 7 from PSL(2, q) and determine all the possible values ofλin the simple 3-(q+1,7,λ) designs admitting PSL(2,q) as an automorphism group.Main Theorem 5:There exists a simple 3-(q+1,7,λ) design with automorphism group PSL(2,q) and 1＜λ≤if and only if one of the following cases holds:(ⅰ) If q= 71,251 (mod 420), thenλ= 0,1,15,21 (mod 35).(ⅱ) If q= 211,391 (mod 420), thenλ= 0,15,21,36 (mod 105).(ⅲ) If q= 3,123,243,303,87,207,387,283,403,103,163,67,187,247,367,19,139, 199,319 (mod 420), then 35|λ.(ⅳ) If q= 31,151,271,331 (mod 420), thenλ= 0,21 (mod 35).(ⅴ) If q= 311,11,131,191 (mod 420), thenλ= 0,21 (mod 105).(ⅵ) If q= 183,363,27,267,43,223,127,307,379,139 (mod 420), thenλ= 0,15 (mod 35).(ⅶ) If q= 323,83,379,239,419 (mod 420), thenλ= 0,15 (mod 105).(ⅷ) If q= 23,143,203,383,47,227,347,59,79,179,299,359 (mod 420), then 105|λ.