Abstract
The Desarguesian, Hall, and Hering translation planes of order q2 are characterized as exactly those translation planes of odd order with spreads in PG (3,q) that admit a linear collineation group with infinite orbits one of length q+1 and i of length (q-q) /i for i=1 or 2.
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Biliotti, M. and Lunardon, G.: Insiemi di derivazione e sottopiani di Baer in un piano di traslazione, Atti. Acc. Naz. Lincei. Cl. Sci. Fis. Mat. Nat. 69(8) (1980), 135-141.
Cameron, P. J.: Finite permutation groups and finite simple groups, Bull. London Math. Soc. 13 (1981), 1-22.
Charnes, Ch.: Quadratic Matrices and the Translation Planes of Order 52, Proc. Marshall Hall Conf., Wiley, New York, 1993.
Czerwinski, T. and Oakden, D.: The translation planes of order twenty-five. J. Combin. Theory Ser. A 59 (1992), 193-217.
Dickson, L. E.: Linear Groups, Dover, New York, 1958.
Foulser, D. A.: Baer p-elements in translation planes, J. Algebra 31 (1974), 354-366.
Foulser, D. A. and Johnson, N. L.: The translation planes of order q 2 that admit SL(2, q) as a collineation group. II. Odd order, J. Geom. 18 (1982), 122-139.
Ganley, M., Jha, V. and Johnson, N. L.: The translation planes admitting a non-solvable doubly transitive line-sized orbit, J. Geom. (to appear).
Hering, Ch.: On shears of translation planes, Abh. Math. Sem. Hamburg 37 (1972), 258-268.
Johnson, N. L.: Translation planes of order q 2 that admit q + 1 elations, Geom. Dedicata 15 (1984), 329-337.
Johnson, N. L.: Translation planes admitting Baer groups and partial flocks of quadric sets, Simon Stevin 63(3) (1989), 163-188.
Liebler, R. A. and Kantor, W. M.: The rank 3 permutation representations of the classical groups, Trans. Amer. Math. Soc. 271 (1982), 1-71.
Lüneburg, H.: Translation Planes, Springer-Verlag, Berlin, 1980.
Ostrom, T. G.: Linear transformations and collineations of translation planes, J. Algebra 14(3) (1970), 405-416.
Ostrom, T. G.: Elations in finite translation planes of characteristic 3, Abh. Math. Sem. Hamburg 41 (1974), 179-184.
Ostrom, T. G.: Translation planes of dimension two with odd characteristic, Canad. J. Math. 32 (1980), 1114-1125.
Payne, S. W. and Thas, J. A.: Conical flocks, partial flocks, derivation, and generalized quadrangles, Geom. Dedicata 38 (1991), 229-243.
Thas, J. A.: Generalized quadrangles and flocks of cones, European J. Combin. 8 (1987), 441-452.
Walker, M.: On translation planes and their collineation groups, Thesis, Univ., London, 1973.
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Hiramine, Y., Jha, V. & Johnson, N.L. Characterization of Translation Planes by Orbit Lengths. Geometriae Dedicata 78, 69–80 (1999). https://doi.org/10.1023/A:1005226621288
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DOI: https://doi.org/10.1023/A:1005226621288