Permutation Group Theory Constructed With A Special Delivery Of The Composite Structure (¦¸, ¦«, ¦©)

This paper aims at discussing the automorphism groups of combinatorial structures (Ω, Λ,I), which are explained as 2-(v, k, 1) designs or graphs respectively when comes to concrete discussions.In Chapter 1, we give a comprehensive survey of the backgrounds and modern developments of the automorphism groups of combinatorial structures (Ω, Λ,I), in which we emphasize the applications of the new theories and new tools developed in abstract groups and permutation groups to the classification of (Ω, Λ.I).In Chapter 2, we explain (Ω, Λ,I) as 2-(v,k, 1) designs and determine the designs admitting a block-primitive automorphism group isomorphic to PSU(3,q²). Part of the work in this chapter is to list all subgroups of PSU(3,q²), which is a key step to determine the corresponding designs. The methods we use can be generalized and then applied to determine the structures of an arbitrary classical group C(n,K) when n is not large. Another work is to prove that any 2-(v,k, 1) design admitting a point-transitive automorphism group isomorphic to PSU(3,q²) is not a projective plane, the results can be generalized to determine whether an almost simple group can act as a point-transitive collineation group of a projective plane or not. The main results in Chapter 2 are the following:Main Theorem 1 Let V be a projective plane and G a collineation group of D. If G is transitive on the points of D, then G is not isomorphic to PSU(3,q²).Main Theorem 2 Let V be a 2-(v, k, 1) design and G an automorphism group of V. If G is primitive on the blocks and G = PSU(3,q²), then V is a Hermitian Unital.In Chapter 3, we explain (Ω, Λ,I) as graphs and discuss the classification of 2-arc transitive graphs admitting an automorphism group isomorphic to the sporadic simple group J₁. The main results in Chapter 3 are the following: Theorem 1 Let G = J₁. Suppose that Γ = (G,T,T_gT) is a (G,2)-arc transitive graph, then up to an isomorphism, T, g and Γ are determined. (They are listed in the table on page 42.)Theorem 2 Let T be one of the graphs in Theorem 1. If G is primitive on the points of Γ, then AutΓ = G; If G is imprimitive on the points of Γ and G 2³ : Z₇, then there exists a simple group N such that G < N and N(?)AutΓ