Each Invertible Sharply d-Transitive Finite Permutation Set withd ≥ 4 is a Group

Abstract

All known finite sharply 4-transitive permutation sets containing the identity are groups, namely i>S4, i>S5, i>A6 and the Mathieu group of degree 11. We prove that a sharply 4-transitive permutation set on 11 elements containing the identity must necessarily be the Mathieu group of degree 11. The proof uses direct counting arguments. It is based on a combinatorial property of the involutions in the Mathieu group of degree 11 (which is established here) and on the uniqueness of the Minkowski planes of order 9 (which had been established before): the validity of both facts relies on computer calculations. A permutation set is said to be invertible if it contains the identity and if whenever it contains a permutation it also contains its inverse. In the geometric structure arising from an invertible permutation set at least one block-symmetry is an automorphism. The above result has the following consequences. i) A sharply 5-transitive permutation set on 12 elements containing the identity is necessarily the Mathieu group of degree 12. ii) There exists no sharply 6-transitive permutation set on 13 elements. For i>d ≥ 6 there exists no invertible sharply i>d-transitive permutation set on a finite set with at least i>d + 3 elements. iii) A finite invertible sharply i>d-transitive permutation set with i>d ≥ 4 is necessarily a group, that is either a symmetric group, an alternating group, the Mathieu group of degree 11 or the Mathieu group of degree 12.