Certain groups with bounded movement having the maximal number of orbits

Let G be a permutation group on a set Omega such that G has no fixed points in Omega. If, for a given positive integer m, the cardinalities \Gamma(g)\Gamma\ is at most m for all g is an element of G and Gamma subset of or equal to Omega, then G is said to have bounded movement m on Omega. When the maximum of \Gamma(g)\Gamma\ over all g is an element of G and Gamma subset of or equal to Omega is equal to m, we say G has bounded movement equal to m. We will show that if G has bounded movement equal to m and p (greater than or equal to 5) is the least odd prime dividing \G\, then it has at most 2m - (p - 1) nontrivial orbits. Moreover the groups G attaining the maximum bound will be classified. (C) 2002 Elsevier Science (USA). All rights reserved.