Limited Local (psl (2, Q) (?) S <sub> 2 </ Sub>, 2) - Arc-transitive

Let r be a finite undirected simple graph with vertex set Vг,an 2-arc ofгis a sequence(v0,v1,v2)of vertices such that vi-1,vi are adjacent for 1≤i≤2 and vo≠v2.A graph r is called(X,2)-arc-transitive,where x≤Autг,if.X is transitive on the set of 2-arcs of r.In particular,r is called locally(X,2)-arc transitive,if for each vertex v,Xv acts transitively on 2-arcs starting at v.The main purpose of this thesis is to characterize finite locally(PSL(2,q)(S2,2)-arc transitive graphs.The main results are the following three theorems.Theorem 1.Let G=T(?)S2,where T is nonabelian simple group.Assume that G is transitive permutation group onΩ=[G:Ga](a∈Ω).LetN=T×T,then one of the followings holds:(1)If Ga=H≮N,then G is quasiprimitive on Q.(2)If Ga=H≤N,then G is bi-quasiprimitive on Q.And(i)If H<T1,then T2 is semiregular on△1=[N:H],T1 is semiregular on△2=[N:Ha].In particular,if H=T1,then T2is regular on△1.If H=T2,then T2 is regular on△2；(ii)IfH≮Ti,(i=1,2),and H=<(t,t)|t∈T>≌T,then N is primitive of日S on△1=[N:H]；(iii)If H=<(t,t)|t∈L>≌L≤T,then T1,T2 are both semiregular on△i(i=1,2)and intransitive.Theorem 2. Let T=PSL(2,q),where q=pn,p is a prime.Assume that∑=Cos(T,L,R)is locally(T,2)-arc transitive and connected graph. Then the follows hold:(1)L≌R≤D2(q±1)/k,where k=(2,q-1),then val(∑)=3； (2)L≌R≌A5,L and R are not conjugate in T,and L∩n R=A4,then val(∑)=5；(3)L≌R≌A5,L and R are not conjugate in T,and L∩R=D10,then val(∑)=6；(4)L≌A4,R≌A5,then val(∑)={1,5}；(5)L≌S4,R≌S4,then val(∑)=4；(6)q=29,L≤D30,R≌A5,then val(∑)={1,6}；(7)q=17,L≤D18,R≌S4,then val(∑)=3；(8)q=8,L≤D18,R≌S4,then val(∑)=4；(9)p=2且L≤Z2n:Z2n-1,then∑≌K2n,2n,val(∑)=2n-1.Theorem 3.LetΓbe bipartite(G,2)-arc transitive graph,where G=T(?)S2,T= PSL(2,q),q=pn,p is a prime.Let N=T×T=T1×T2,△1,△2 are the biparts ofΓ.Then(1)Γ=Kn,n,where n=|△i|.(2)Γis a multiple normal cover of∑,where∑is locally(T,2)-arc transi-tive,thus satisfies Theorem 2.