Holomorph Type Primitive Permutation Group Of 2-closure,

Let G be a transitive permutation group onΩ. Let Orbl(G,Ω) be the set of orbitals of G onΩ, That is:the set of orbits of the naturally induced action of G onΩ:=ΩxΩ. Then G(2):={x∈Sym(Ω)|Ox=O,(?)O∈Orbl(G,Ω)} is called 2—Closure of G. Thus the 2—Closure of G is the largest subgroup of Sym(Ω) which fixes Orbl(G,Ω) pointwise.Let G≤Sym(Q) be a transitive permutation group on a setΩ. If every nontrivial normal subgroup N of G is transitive onΩ, then G is called quasi-primitive. According to the O'Nan—Scott theorem, G is divided into 8 types—of type HS,HC,HA,AS,SD,CD,TW,PA, in which of types HS,HC,HA are Holomorph type. However, this thesis discuss mainly that G is of type HS or of type HC, and the main results are the following two theorems:Theorem 1. Suppose T is a non-abelian simple group. Let G= T. Aut(T) be a primitive permutation group of type HS. Then the following statements hold:(i) If there exists t∈T which is not conjugate to t-1 in Aut(T), then G(2)=G.(ii) If each t∈T is conjugate to t-1 in Aut(T), then G(2)= G.2.Theorem 2. Suppose T is a non-abelian simple group and N= Td with (d≥2). Let G= N. Aut(N)be a primitive permutation group of type HC, Then the following statements hold:(i) If there exists t∈T which is not conjugate to t-1 in Aut(T), then G(2)= G. (ii) If each t∈T is conjugate to-1 in Aut(T), thenG(2)= [T2.(Out(T)x S2)]lSd.A Graphгis called arc-transitive if Aut(г) is transitive on the arc-set ofг. The following theorem is a result about the arc-transitive graphs.Theorem 3. Let G be a group and X= G:Aut(G)< Sym(G). Then the following statements hold:(i) Suppose a graphгis X-edge transitive Cayley graph of G, thenгis arc-transitive graph.(ii) If G is abelian, thenгis X-arc-transitive; If G is non-abelian, then Aut{г)≥X.2.Corollary 1. Letг= Cay(G, S) be a normal edge-transitive Cayley graph of X on G. Suppose Inn(G)≤X1, thenгis arc-transitive graph.