Each Invertible Sharply d-Transitive Finite Permutation Set with d \geq 4 is a Group
Arrigo Bonisoli
and Pasquale Quattrocchi
DOI: 10.1023/A:1011211907282
Abstract
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 d 
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 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.
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 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.Pages: 241–250
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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References
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2. W. Benz, “Permutations and plane sections of a ruled quadric,” Symposia Mathematica, Istituto Nazionale di Alta Matematica 5 (1970), 325-339.
3. W. Benz, Vorlesungen \ddot uber Geometrie der Algebren, Springer, Berlin, 1973.
4. A. Bonisoli and T. Grundh\ddot ofer, “On the uniqueness of the Minkowski 2-structure of order 9 possessing a reflection,” Research Report, Modena, November 1987.
5. J.H. Conway, “M13,” in Surveys in Combinatorics, 1997, R.A. Bailey (Ed.), Cambridge Univ. Press, Cambridge, 1997, pp. 1-11.
6. M. Hall, The Theory of Groups, Macmillan, New York, 1959.
7. W. Heise and H. Karzel, “Laguerreund Minkowski-m-Strukturen,” Rend. Ist. Matem. Univ. Trieste 4 (1972), 139-147.
8. W. Heise and K. S\ddot orensen, “Scharf n-fach transitive Permutationsmengen,” Abh. Math. Sem. Univ. Hamburg 43 (1975), 144-145.
9. B. Huppert and N. Blackburn, Finite Groups III, Springer, Berlin 1982.
10. M.J. Kallaher, “On finite Bol quasifields,” Algebras, Groups and Geometries 3 (1985), 300-312.
11. C.W.H. Lam, G. Kolesowa, and L. Thiel, “A computer search for finite projective planes of order 9,” Discrete Math. 92 (1991), 187-195.
12. P. Lancellotti, “Una nuova classe di insiemi di permutazioni strettamente n-transitivi,” Atti Sem. Mat. Fis. Univ. Modena 30 (1981), 83-93.
13. A. Pasini, “Diagram geometries for sharply n-transitive sets of permutations or of mappings,” Des. Codes Cryptography 1 (1992), 275-297.
14. N. Percsy, “Finite Minkowski planes in which every circle-symmetry is an automorphism,” Geom. Dedicata 10 (1981), 269-282.
15. B. Polster, “Invertible sharply n-transitive sets,” J. of Combinatorial Th., Ser. A 81 (1998), 231-254.
16. P. Quattrocchi and C. Fiori, “A result concerning the existence of certain Minkowski-2-structures,” J. of Geometry 14 (1980), 139-142.
17. P. Quattrocchi and G. Rinaldi, “Insiemi di permutazioni strettamente 4-transitivi e gruppo di Mathieu M11,” Boll. Un. Mat. Ital. 11B (7), (1997), 319-325.
18. B. Segre, Istituzioni di Geometria Superiore, Vol. III, Istituto Matematico `G. Castelnuovo,' Roma, 1965.
19. G.F. Steinke, “A remark on Benz planes of order 9,” Ars Combinatoria 34 (1992), 257-267.