ON PERMUTATION-GROUPS WITH BOUNDED MOVEMENT

Let 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{lunate} 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. © 1991.