Innovations in Incidence Geometry

Transitive eggs

John Bamberg and Tim Penttila

Full-text: Open access


We prove that a pseudo-oval or pseudo-ovoid (that is not an oval or ovoid) admitting an insoluble transitive group of collineations is elementary and arises over an extension field from a conic, an elliptic quadric, or a Suzuki-Tits ovoid.

Article information

Innov. Incidence Geom., Volume 4, Number 1 (2006), 1-12.

Received: 20 December 2005
Accepted: 6 July 2006
First available in Project Euclid: 28 February 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 51E20: Combinatorial structures in finite projective spaces [See also 05Bxx]

pseudo-oval pseudo-ovoid egg translation generalised quadrangle transitive


Bamberg, John; Penttila, Tim. Transitive eggs. Innov. Incidence Geom. 4 (2006), no. 1, 1--12. doi:10.2140/iig.2006.4.1.

Export citation


  • V. Abatangelo and B. Larato, A characterization of Denniston's maximal arcs, Geom. Dedicata 30 (1989), no. 2, 197–203.
  • B. Bagchi and N. S. N. Sastry, Even order inversive planes, generalized quadrangles and codes, Geom. Dedicata 22 (1987), no. 2, 137–147.
  • J. Bamberg and T. Penttila, A classification of transitive ovoids, spreads, and $m$-systems of polar spaces, preprint.
  • ––––, Overgroups of cyclic sylow subgroups of linear groups, UWA Research Report 2005/09.
  • A. Barlotti, Un'estensione del teorema di Segre-Kustaanheimo, Boll. Un. Mat. Ital. (3) 10 (1955), 498–506.
  • M. R. Brown and M. Lavrauw, Eggs in ${\rm PG}(4n-1,q)$, $q$ even, containing a pseudo-pointed conic, European J. Combin. 26 (2005), no. 1, 117–128.
  • F. Buekenhout, A. Delandtsheer, J. Doyen, P. B. Kleidman, M. W. Liebeck, and J. Saxl, Linear spaces with flag-transitive automorphism groups, Geom. Dedicata 36 (1990), no. 1, 89–94.
  • B. N. Cooperstein, Minimal degree for a permutation representation of a classical group, Israel J. Math. 30 (1978), no. 3, 213–235.
  • A. Cossidente and O. H. King, Group-theoretic characterizations of classical ovoids, In Finite geometries, volume 3 of Dev. Math., 121–131. Kluwer Acad. Publ., Dordrecht, 2001.
  • J. D. Dixon and B. Mortimer, Permutation groups, Springer-Verlag, New York, 1996.
  • The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.4, 2005. \verb+(
  • R. Guralnick, T. Penttila, C. E. Praeger, and J. Saxl, Linear groups with orders having certain large prime divisors, Proc. London Math. Soc. (3) 78 (1999), no. 1, 167–214.
  • C. Jansen, K. Lux, R. Parker, and R. Wilson, An atlas of Brauer characters, London Mathematical Society Monographs, New Series 11, The Clarendon Press Oxford University Press, New York, 1995. Appendix 2 by T. Breuer and S. Norton, Oxford Science Publications.
  • P. Kleidman and M. Liebeck, The subgroup structure of the finite classical groups, London Mathematical Society Lecture Note Series 129, Cambridge University Press, Cambridge, 1990.
  • H. Lüneburg, Die Suzukigruppen und ihre Geometrien, Springer-Verlag, Berlin, 1965.
  • ––––, Translation planes, Springer-Verlag, Berlin, 1980.
  • H. H. Mitchell, Determination of the ordinary and modular ternary linear groups, Trans. Amer. Math. Soc. 12 (1911), no. 2, 207–242.
  • C. M. O'Keefe and T. Penttila, Symmetries of arcs, J. Combin. Theory Ser. A 66 (1994), no. 1, 53–67.
  • G. Panella, Caratterizzazione delle quadriche di uno spazio (tridimensionale) lineare sopra un corpo finito, Boll. Un. Mat. Ital. (3) 10 (1955), 507–513.
  • S. E. Payne and J. A. Thas, Finite generalized quadrangles, Research Notes in Mathematics 110, Pitman (Advanced Publishing Program), Boston, MA, 1984.
  • T. Penttila, Translation generalised quadrangles and elation laguerre planes of order 16, European J. Combin., to appear.
  • B. Segre, Sulle ovali nei piani lineari finiti, Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8) 17 (1954), 141–142.
  • M. Suzuki, On a class of doubly transitive groups, Ann. of Math. (2) 75 (1962), 105–145.
  • J. A. Thas and K. Thas, Translation generalized quadrangles in even characteristic, Combinatorica 26 (2006), 709–732.
  • K. Zsigmondy, Zur theorie der potenzreste, Monatsh. für Math. u. Phys. 3 (1892), 265–284.