The The Cubic Free Times Quasiprimitive And Two Quasiprimitive Permutation Group

Primitive permutation groups have received much attention in the literature. Primitive permutation groups of odd degree are determined in [11], and primitive permutation groups of prime power degree are determined in [4]. More recently, the list of primitive permutation groups of squarefree degree is given in [9]. These char-acterizations have been successfully used in characterizing some classes of symmetric graphs of odd order[7], of prime power order[8], and of squarefree order[10].A permutation group G ofΩis a subgroup of the symmetric group sym(Ω), and the size|Ω|is called the degree of G. A permutation group G on a setΩis called quasiprimitive if each non-trivial normal subgroup of G is transitive onΩ. G is called bi-quasiprimitive if each non-trival normal subgroup of G has at most two orbits, and there exists at least one normal subgroup of G which has exactly two orbits. And G is called of cube-free degree if there is no prime p such that|Ω| is divisible by p3.The main purpose of this thesis is along the schedule of O'Nan-Scott Theo-rem, giving characterizations of quasiprimitive and bi-quasiprimitive permutation groups of cube-free degree. The main results are the following two theorems.Theorem 1. Let G≤Sym(Ω) be a quasiprimitive permutation group of cube-free degree. Then either G is almost simple type, or one of the following holes.(1) G is holomorph affine type, and one of the following holds, wherel| p—1.(2) soc(G)= Zp2, and one of the following holds: with l divisible by an odd divisor of p+1; with l≠1 dividing p—1; (e)SL(2,5)≤Gw≤Zp-1.SL(2,5).2,where 5 divides p2-1.(ii)G is holomorph simple type,and soc(G)=PSL(2,p2)with p≡3(mod8).(iii)G is simple diagonal type,and N=soc(G)=T2.,G=N.O,where T=PSL(2,p),Z2≤O≤Out(T)×S2,and p is a prime such that p≡土3(mod8).(iv)G is product action type,Then soc(G)=T2,where T is a nonabelian simple group and has a subgroup of index square free.Theorem 2.Let G be a bi-quasiprimitiVe permutation groupp onΩwith bi-partition△1 and△2,where |Ω|is cube.free.Let G+=G△1:G△2.Then further,one of the following hold.(1)If G+is faithful on△1,then G+is quasiprimitive,and G+is AS,HA,PA,and G+is as in Theorem 1.(2)If G+is unfaithful on△1,then(G+)△1 is quasiprimitive and(G+)△1 is AS,HA or P4,and G+=(N1×N2).(O1.O2),where N1≌N2=Soc((G+)△1).