Almost Simple Groups And Combinatorial Designs

This paper aims at discussing the automorphism groups of combinatorial design S=(P,L).In Chapter 1, we give a comprehensive survey of the backgrounds and modern developments of the automorphism groups of combinatorial design S = (P,L). In Chapter 2, we introduce the elementary concepts that will be used in this thesis.In Chapter 3 and Chapter 4, we discuss almost simple groups and line-transitive spaces: Let S = (P,L), G≤Aut(S), and T(?)G≤Aut(T), where T is a finite simple groups. The main results are the following:Main theorem 1. Let T(?)G3D₄(q), then T is line-transitive on S, where q is a power of p, p is a prime.Main theorem 2. Let T≌Sz(q)(?)G≤Aut(T) and G acts line-transitively and point-primitively on a finite linear space S=(P,L). Suppose that T is point-transitive but not line-transitive on S, then G_α∩T≌Z_q+(?)+1:Z₄ and 8(?)|G_L∩T|, where q=2^α,α>1 is odd, t² =2q andε=±. Main theorem 3. Let T≌PSU(3,q)(?)G≤Aut(PSU(3,q)) and G acts line-transitively on a finite linear space S = (P,L). then one of the following holds:(1) S = PG(2,q) , the Desarguesian projective plane with parameters(b,v,r,k) = (q²(q² -q +1),q³ + 1,q² -q+1,q-1), is a Hermitian unitary design;(2) PSU(3,q) is point-transitive but not line-transitive on S. Furthermore, T_α=PSU(3,q₀), where q=q₀^αfor some integerα.(3) PSU(3,q) is line-transitive but not flag-transitive on S, then T_α=(Z_q+1/3×Z_q+1):S₃.In Chapter 5 and Chapter 6, we discuss almost simple groups and flag-transitive spaces: The main results are the following:Main theorem 4. Let S = (P,L) is a non-trivial 5-(v,k,1) design. Suppose that T≌PSL(2,q)≤G≤Aut(T) acts flag-transitively on S, then G≌PSL(2,23) and S is 5-(24,8,1) design.Main theorem 5. There are not non-trivial 6-(v,k,2) flag-transitive designs.