Let G be a transitive permutation group on a set ? and let m be a positive integer.
If no element of G moves any subset of ? by more than m points, then |? | [2mp I
(p-1)] wherep is the least odd primedividing |G |. When the bound is attained, we show that | ? | = 2 p q ….. q where ? is a non-negative integer with 2 < p, r 1 and q is a prime satisfying p < q < 2p, ? = 0 or 1, I i n. Furthermore, every 2-element of G fixes at least [2m/(p- 1)] points and each q -element of G fixes at least [2m(q -p)/(p- 1)( q - l)] points. Finally, we prove that if G is a p-group of exponent, at least p2 and I ? l = [2mp /(p- l)], then every fixed point free element of G has order p.