Transitive Permutation Groups with Bounded Movement Original Research Article

Suppose a transitive permutation group G on Ω is such that for each g set membership, variant G, Δ subset of or equal to Ω, if Δ ∩ Δg = φ then Δ ≤ m. Then, by [1], Ω ≤ 3m. The bound is sharp. The few known examples where the bound is attained are (i) G = S3, m = 1; (ii) G = A4, A5, m = 2; (iii) G is a 3 group, m = 3r. We conjecture that this list is complete, that is, that the groups for which the bound is sharp are essentially finite 3-groups. We show that a minimal counterexample to this conjecture must be a primitive simple group.