On permutation groups with bounded movement

AbstractLet G be a permutation group on a set Ω with no fixed points in Ω and let m be a positive integer. If no element of G moves any subset of Ω by more than m points (that is if ¦Γ − Γg¦⩽ m for every Γ ⊆ Ω and g ϵ G), then Ω is finite, and moreover ¦Ω¦$̌5m − 2. If moreover G is transitive on Ω then ¦Ω¦$̌3m and equality can be achieved if m = 2 or m is a power of 3