Innovations in Incidence Geometry

A geometric proof of Wilbrink's characterization of even order classical unitals

Alice M. W. Hui

Full-text: Open access


Using geometric methods and without invoking deep results from group theory, we prove that a classical unital of even order n4 is characterized by two conditions (I) and (II): (I) is the absence of O’Nan configurations of four distinct lines intersecting in exactly six distinct points; (II) is a notion of parallelism. This was previously proven by Wilbrink (1983), where the proof depends on the classification of finite groups with a split BN-pair of rank 1.

Article information

Innov. Incidence Geom., Volume 15, Number 1 (2017), 145-167.

Received: 13 January 2015
Accepted: 24 February 2015
First available in Project Euclid: 28 February 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05B25: Finite geometries [See also 51D20, 51Exx] 05B25: Finite geometries [See also 51D20, 51Exx] 51E20: Combinatorial structures in finite projective spaces [See also 05Bxx] 51E21: Blocking sets, ovals, k-arcs 51E23: Spreads and packing problems

unital classical unital Hermitian curve spread


Hui, Alice M. W. A geometric proof of Wilbrink's characterization of even order classical unitals. Innov. Incidence Geom. 15 (2017), no. 1, 145--167. doi:10.2140/iig.2017.15.145.

Export citation


  • J. André, Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe, Math. Z. 60 (1954), 156–186.
  • R. D. Baker, G. L. Ebert, G. Korchmáros and T. Szőnyi, Orthogonally divergent spreads of Hermitian curves, in: F. de Clerck, J. Hirschfeld (Eds.), Finite Geometry and Combinatorics in: London Math. Soc. Lecture Note Ser., vol.13, Cambridge Univ. Press, Cambridge, 1993, pp.17–30.
  • S. G. Barwick, A characterization of the classical unital, Geom. Dedicata 52 (1994), 175–180.
  • S. G. Barwick and G. Ebert, Unitals in Projective Planes, Springer Monogr. Math., Springer-Verlag, New York, NY, 2008.
  • R. H. Bruck, Construction problems of finite projective planes, in R. C. Bose, T. A. Dowling (Eds.), Combinatorial mathematics and its applications, Chapter 27, Univ. of North Carolina Press, Chapel Hill, 1969, pp. 426–514.
  • R. H. Bruck and R. C. Bose, The construction of translation planes from projective spaces, J. Algebra 1 (1964), 85–102.
  • ––––, Linear representations of projective planes in projective spaces, J. Algebra 4 (1966), 117–172.
  • A. A. Bruen and J. W. P. Hirschfeld. Intersections in Projective Space II: Pencils of Quadrics. European J. Combin. 9 (1988), 255–270.
  • F. Buekenhout, Existence of unitals in finite translation planes of order $q^2$ with a kernel of order $q$, Geom. Dedicata 5 (1976), 189–194.
  • P. J. Cameron and N. Knarr, Tubes in $\mbox{\rm PG}(3, q)$, European J. Combin. 27 (2006), 114–124.
  • P. Dembowski, Inversive planes of even order, Bull. Amer. Math. Soc. 69 (1963), 850–854.
  • ––––, Finite Geometries, Springer-Verlag, New York, NY, 1968.
  • P. Dembowski and D. R. Hughes, On finite inversive planes, J. London Math. Soc. 40 (1965), 171–182.
  • J. Dover, Subregular spreads of Hermitian unitals, Des. Codes Cryptogr. 39 (2006), 5–15.
  • J. W. P. Hirschfeld, Projective Geometries over Finite Fields, second ed., Clarendon Press, Oxford, 1998.
  • ––––, Finite Projective Spaces of Three Dimensions, Oxford Univ. Press, Oxford, 1985.
  • D. R. Hughes and F. C. Piper, Projective Planes, GTM, vol. 6, Springer-Verlag, Berlin, 1973.
  • A. M. W. Hui and P. P. W. Wong, On embedding a unitary block design as a polar unital and an intrinsic characterization of the classical unital, J. Combin. Theory Ser. A 122 (2014), 39–52.
  • A. M. W. Hui, Extending some induced substructures of an inversive plane, Des. Codes Cryptogr. 79 (2016), 611–617.
  • D. Luyckx, A geometric construction of the hyperbolic fibrations associated with a flock, $q$ even, Des. Codes Cryptogr. 39 (2006), 281–288.
  • M. E. O'Nan, Automorphisms of unitary block designs, J. Algebra 20 (1972), 495–511.
  • S. E. Payne and J. A. Thas, Finite Generalized Quadrangles, second ed., EMS Ser. Lect. Math., European Mathematical Society (EMS), Zürich, 2009.
  • F. C. Piper, Unitary block designs, in: R. J. Wilson (Ed.), Graph Theory and Combinatorics, in: Res. Notes Math., vol. 34, Pitman, Boston, 1981, pp. 98–105.
  • J. A. Thas, Flocks of finite egglike inversive planes, in: A. Barlotti (Ed.), Finite Geometric Structures and their Applications. C.I.M.E., II Ciclo, Bressanone, Edizioni Cremonese, Rome, 1972, pp. 189–191.
  • H. Wilbrink, A characterization of the classical unitals, in: N. L. Johnson, M. J. Kallahar, C. T. Long (Eds.), Finite Geometries, in: Lect. Notes Pure Appl. Math., vol. 82, Marcel Dekker, New York, 1983, pp. 445–454.