For a permutation group G on a set S, the moยจement of G is defined as the maximum cardinality of subsets T of S for which there exists an element x g G x ลฝ such that T is disjoint from its translate T that is, when such subsets have . bounded cardinality . It was shown by the second author that, if G has bounded < < movement m and if G has no fixed points in S, then S is finite, and S is bounded < < above by a function of m. In particular, if G is transitive, then S F 3m. This paper completes the proof of a conjecture of Gardiner and Praeger that the only transitive groups on a set of size 3m which have movement m are transitive ลฝ . permutation groups of exponent 3 when m is a power of 3 , the symmetric group S in its natural representation on a set of three points, and the alternating groups 3 A and A , in their transitive representations on six points. แฎ 1996 Academic Press, 4 5
Inc.
Let G be a permutation group on a set S, and consider the family of subsets T of S for which there exists an element x g G such that T is disjoint from its translate T x . If the cardinality of such subsets T is bounded, then we say that G has bounded moยจement and we define the moยจement of G as the maximal cardinality of such a subset T of S. This w x ลฝ w x. notion we introduced in Pr or see GP and a fuller discussion of * Some of the work reported in this paper was done during the Oberwolfach meeting on permutation groups, 16แ22 January, 1994.