Abstract
We classify the pairs (S, G) where S is a finite n-dimensional linear space with n ≥ 4 and G is an automorphism group of S acting transitively on the (line, hyperplane)-flags. We show in particular that S must be either a Desarguesian projective or affine space provided with its subspaces of dimension ≤ n - 1, or a Mathieu-Witt design provided with its blocks and its subsets of size ≤ n - 1. Our proof uses a recent classification of the flag transitive 2-(v, k, 1) designs, which in turn heavily depends on the classification of all finite simple groups. The case n = 3 has been settled in another paper.
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References
Birkhoff, G. 1967. Lattice Theory, (3rd ed.), Providencs: A.M.S. Colloq. Publ.
Buekenhout, F. 1979. Diagrams for geometries and groups. J. Combin. Theory Ser. A, 27: 121–151.
Buekenhout, F., Delandtsheer, A. and Doyen, J. 1988. Finite linear spaces with flag-transitive groups. J. Combin. Theory Ser. A, 49:268–293.
Buekenhout, F., Delandtsheer, A., Doyen, J., Kleidman, P., Liebeck, M., and Saxl J. 1990. Linear spaces with flag-transitive automorphism groups. Geom. Dedicata 36:89–94.
Delandtsheer, A. (forthcoming). Finite (line, plane)-flag transitive planar spaces.
Huppert, B. 1967. Endliche Gruppen I. Berlin: Springer-Verlag.
Kantor, W.M. 1972. k-homogeneous groups. Math. Z. 124:261–265.
Kantor, W.M. 1985. Homogeneous designs and geometric lattices. J. Combin. Theory Ser. A, 38:66–74.
Kung, J.P.S. 1979. The Radon transform of a combinatorial geometry I, J. Combin. Theory Ser. A, 26:97–102.
Welsh, D.J.A. 1976. Matroid Theory, London: Academic Press.
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Communicated by D. Jungnickel
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Delandtsheer, A. Dimensional linear spaces whose automorphism group is (line, hyperplane)-flag transitive. Des Codes Crypt 1, 237–245 (1991). https://doi.org/10.1007/BF00123763
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DOI: https://doi.org/10.1007/BF00123763