TRANSITIVE PERMUTATION-GROUPS WITH BOUNDED MOVEMENT

Suppose a transitive permutation group G on Omega is such that for each g is an element of G, Delta subset of or equal to Omega, if Delta boolean AND Delta(g) = phi then \Delta\ less than or equal to m. Then, by [1], \Omega\ less than or equal to 3m. The bound is sharp. The few known examples where the bound is attained are (i) G = S-3, m = 1; (ii) G = A(4), A(5), m = 2; (iii) G is a 3 group, m = 3(r). 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. (C) 1994 Academic Press, Inc.