Transitive Permutation Groups of Minimal Movement Original Research Article

For a permutation groupGon a setS, themovementofGis defined as the maximum cardinality of subsetsTofSfor which there exists an elementxset membership, variantGsuch thatTis disjoint from its translateTx(that is, when such subsets have bounded cardinality). It was shown by the second author that, ifGhas bounded movementmand ifGhas no fixed points inS, thenSis finite, and S is bounded above by a function ofm. In particular, ifGis transitive, then S≤3m. This paper completes the proof of a conjecture of Gardiner and Praeger that the only transitive groups on a set of size 3mwhich have movementmare transitive permutation groups of exponent 3 (whenmis a power of 3), the symmetric groupS3in its natural representation on a set of three points, and the alternating groupsA4andA5, in their transitive representations on six points.