Geodesic
Each invertible sharply $d$-transitive finite permutation set with $d\geq 4$ is a group
Journal of Algebraic Combinatorics, Tome 12 (2000) no. 3, pp. 241-250.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: All known finite sharply 4-transitive permutation sets containing the identity are groups, namely $S _{4}, S _{5}, A _{6}$ 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 $dge$ 6 there exists no invertible sharply $d$-transitive permutation set on a finite set with at least $d + 3$ elements. iii) A finite invertible sharply $d$-transitive permutation set with $dge$ 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.
Classification : transitive, permutation, sets, for, d, geq, 4:, each, such, permutation, set, turns, out, to, be, a, group,, hence, either, a, symmetric, group,, an, alternating, group,, the, Mathieu, group, of, degree, 11, or, the, Mathieu, group, of, degree, 12,, see, Proposition, 12
Keywords: sharply $d$-transitive permutation set, Mathieu groups of degrees 11 and $12$, (B)$-geometry arising from a permutation set$, block-symmetry 
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     title = {Each invertible sharply $d$-transitive finite permutation set with $d\geq 4$ is a group},
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Bonisoli, Arrigo; Quattrocchi, Pasquale. Each invertible sharply $d$-transitive finite permutation set with $d\geq 4$ is a group. Journal of Algebraic Combinatorics, Tome 12 (2000) no. 3, pp. 241-250. http://geodesic.mathdoc.fr/item/JAC_2000__12_3_a3/