Two Problems On The Movement Of Permutation Groups

Bounded movement is a concept of permutation group which was introduced by Professor Praeger in[15].Let G be a permutation group on a setΩwith no fixed points inΩ.If the cardinalities |Γ^g\Γ| are bounded for all g∈G andΓ(?)Ω,then G is said to have bounded movement.The movement of G is defined as move(g) =(?)|Γ^g\Γ|.In particular,if all nonidentity elements of G have the same movement,then we say that G has constant movement.If G has bounded movement onΩ,Praeger proved that |Ω| is finite and gave a boundary of |Ω|.Meanwhile,she obtained the boundary of G-orbits' length and number in[15]. After that,groups with bounded movement have interested many scholars.For example,A. Gardiner,A.Mann,A.Hassani,M.Khayaty,E.I.Khukhro,J.R.Cho,P.S.Kim,P.M. Neumann,M.Alaeiyan,S.Yoshiara and so on.They have done lots of work(see references therein).The boundary which were given by Praeger in[15]have been improved.They have obtained some different boundaries,and classified groups when the boundaries are obtained. Suppose that |Ω| obtain the boundary exactly which was given in[15],Alaeiyan classified all transitive permutation groups with a constant movement.We extend Alaeiyan's work in[2] from two aspects in this paper.The main results are the following:(1) We classify all permutation groups with a constant movement without the supposition that |Ω| obtain the boundary exactly which was given in[15].(2) Suppose that the movement of non-identity elements of a group equal to m₁ or equal to m₂,we obtain elements' order.Moreover,we classify transitive permutation groups with the supposition that |Ω| obtain the boundary exactly which was given in[15].