Permutation groups with bounded movement having maximum orbits

Let G be a permutation group on a set Omega with no fixed points in Omega and let m be a positive integer. If no element of G moves any subset of Omega by more than m points (that is, |I" (g) a-aEuro parts per thousand I"| a parts per thousand currency signaEuro parts per thousand m for every and g aaEuro parts per thousand G), and also if each G-orbit has size greater than 2, then the number t of G-orbits in Omega is at most . Moreover, the equality holds if and only if G is an elementary abelian 3-group.