Improvement on the bounds of permutation groups with bounded movement

Let G be a permutation group on a set 11 with no fixed points in Q and let m be a positive integer. Then we define the movement of G as, m := move(G) := sup(Gamma){\Gamma(g) \ Gamma\ \ g is an element of G}. Let p be a prime, p greater than or equal to 5. If G is not a 2-group and p is the least odd prime dividing \G\, then we show that n := \Omega\ less than or equal to 4m - p + 3. Moreover, if we suppose that the permutation group induced by G on each orbit is not a 2-group then we improve the last bound of n and for an infinite family of groups the bound is attained.