Unitals with many involutory translations

Theo Grundhöfer1, Markus J. Stroppel2, Hendrik Van Maldeghem3
1Institut für Mathematik, Universität Würzburg, Würzburg, Germany
2LExMath, Universität Stuttgart, Stuttgart, Germany
3Department of Mathematics, Ghent University, Gent, Belgium

Tóm tắt

If every point of a unital is fixed by a non-trivial translation and at least one translation has order two then the unital is classical (i.e., hermitian).

Tài liệu tham khảo

Al-Azemi, A., Betten, A., Betten, D. (2014) Unital designs with blocking sets. Discrete Appl. Math. 163(part 2):102–112. https://doi.org/10.1016/j.dam.2013.02.023 Aschbacher, M.: A condition for the existence of a strongly embedded subgroup. Proc. Am. Math. Soc. 38, 509–511 (1973). https://doi.org/10.2307/2038941 Barwick, S.G., Ebert, G.: Springer Monographs in Mathematics, Unitals in Projective Planes (2008). https://doi.org/10.1007/978-0-387-76366-8 Coulter, R.S., Matthews, R.W.: Planar functions and planes of Lenz–Barlotti class II. Des. Codes Cryptogr. 10(2), 167–184 (1997). https://doi.org/10.1023/A:1008292303803 de Resmini, M.J., Hamilton, N.: Hyperovals and unitals in Figueroa planes. Eur. J. Combin. 19(2), 215–220 (1998). https://doi.org/10.1006/eujc.1997.0166 Dembowski, P.: Finite geometries, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol 44. Springer-Verlag, Berlin (1968). https://doi.org/10.1007/978-3-642-62012-6 Dempwolff, U.: \({\rm PSL}(3, q)\) on projective planes of order \(q^3\). Geom. Dedicata 18(1), 101–112 (1985a). https://doi.org/10.1007/BF00221208 Dempwolff, U.: (1985b) On the automorphism group of planes of Figueroa type. Rend. Sem. Mat. Univ. Padova 74:59–62. http://www.numdam.org/item?id=RSMUP_1985__74__59_0 Figueroa, R.: A family of not \((V,\, l)\)-transitive projective planes of order \(q^{3}\), \(q{\lnot \equiv }1~(mod \; 3)\) and \(q>2\). Math. Z. 181(4), 471–479 (1982). https://doi.org/10.1007/BF01182385 Gorenstein, D.: Finite groups, 2nd edn. Chelsea Publishing Co., New York (1980) Grundhöfer, T.: A synthetic construction of the Figueroa planes. J. Geom. 26(2), 191–201 (1986). https://doi.org/10.1007/BF01227843 Grüning, K.: A class of unitals of order \(q\) which can be embedded in two different planes of order \(q^2\). J. Geom. 29(1), 61–77 (1987). https://doi.org/10.1007/BF01234988 Grundhöfer, T., Krinn, B., Stroppel, M.J.: Non-existence of isomorphisms between certain unitals. Des. Codes Cryptogr. 60(2), 197–201 (2011). https://doi.org/10.1007/s10623-010-9428-2 Grundhöfer, T., Stroppel, M.J., Van Maldeghem, H.: Unitals admitting all translations. J. Combin. Des. 21(10), 419–431 (2013). https://doi.org/10.1002/jcd.21329 Grundhöfer, T., Stroppel, M.J., Van Maldeghem, H.: A non-classical unital of order four with many translations. Discrete Math. 339(12), 2987–2993 (2016). https://doi.org/10.1016/j.disc.2016.06.008 Grundhöfer, T., Stroppel, MJ., Van Maldeghem, H.: (2021-11-01) Finite subunitals of the hermitian unitals J. Geom., to appear (2021a). arxiv:2201.02701 Grundhöfer, T., Stroppel, MJ., Van Maldeghem, H.: Moufang sets generated by translations in unitals. J. Combin. Des. 30(2), 91–104 (2021b). https://doi.org/10.1002/jcd.21813 (2021b). arxiv:2008.11445 Hering, C.: On subgroups with trivial normalizer intersection. J. Algebra 20, 622–629 (1972). https://doi.org/10.1016/0021-8693(72)90075-0 Hering, C.: On projective planes of type VI. In: Colloquio Internazionale sulle Teorie Combinatorie (Rome, 1973), Tomo II. Accademia Nazionale dei Lincei, Rome, p 29–53, tenuto a Roma, 3–15 settembre 1973, Atti dei Convegni Lincei, No. 17 (1976) Hering, C., Schaeffer, HJ.: On the new projective planes of R. Figueroa. In: Combinatorial theory (Schloss Rauischholzhausen, 1982), Lecture Notes in Math., vol 969. Springer, Berlin-New York, p 187–190 (1982). https://doi.org/10.1007/BFb0062982 Higman, D.G., McLaughlin, J.E.: Rank \(3\) subgroups of finite symplectic and unitary groups. J. Reine. Angew. Math. 218, 174–189 (1965). https://doi.org/10.1515/crll.1965.218.174 Hughes, D.R., Piper, F.C.: Projective Planes, Graduate Texts in Mathematics, vol. 6. Springer, New York (1973) Hui, A.M.W., Wong, P.P.W.: Non-classical polar unitals in finite Figueroa planes. J. Geom. 103(2), 263–273 (2012). https://doi.org/10.1007/s00022-012-0121-7 Hui, A.M.W., Law, H.F., Tai, Y.K., et al.: Non-classical polar unitals in finite Dickson semifield planes. J. Geom. 104(3), 469–493 (2013). https://doi.org/10.1007/s00022-013-0174-2 Kantor, W.M.: Homogeneous designs and geometric lattices. J. Combin. Theory Ser. A 38(1), 66–74 (1985). https://doi.org/10.1016/0097-3165(85)90022-6 Knarr, N., Stroppel, M.J.: Polarities and unitals in the Coulter–Matthews planes. Des. Codes Cryptogr. 55(1), 9–18 (2010). https://doi.org/10.1007/s10623-009-9326-7 Krčadinac, V., Vlahović, R.: New quasi-symmetric designs by the Kramer–Mesner method. Discrete Math. 339(12), 2884–2890 (2016). https://doi.org/10.1016/j.disc.2016.05.030 Krčadinac, V., Nakić, A., Pavčević, M.O.: The Kramer–Mesner method with tactical decompositions: some new unitals on 65 points. J. Combin. Des. 19(4), 290–303 (2011). https://doi.org/10.1002/jcd.20277 Lüneburg, H.: Translation Planes. Springer, Berlin (1980) Möhler, V. Automorphisms of (affine) SL(2,q)-unitals. Innov. Incid. Geom. to appear (2021a) . arxiv:2012.10116 Möhler, V.: Parallelisms and translations of (affine) SL(2, q)-unitals. J. Geom. 112, 44 (2021b). https://doi.org/10.1007/s00022-021-00611-5 Möhler, V.: Three affine SL(2,8)-unitals. Beitr. Algebra Geom. to appear (2021c). arxiv:2012.10134 O’Nan, M.E.: Automorphisms of unitary block designs. J. Algebra 20, 495–511 (1972). https://doi.org/10.1016/0021-8693(72)90070-1 Tai, Y.K., Wong, P.P.W.: On the structure of the Figueroa unital and the existence of O’Nan configurations. Discrete Math. 330, 41–50 (2014). https://doi.org/10.1016/j.disc.2014.04.012 Taylor, D.E.: The geometry of the classical groups, Sigma Series in Pure Mathematics, vol. 9. Heldermann, Berlin (1992)