Primitive collineation groups of ovals with a fixed point

  • Authors:

  • Arrigo Bonisoli;Gloria Rinaldi

  • Affiliations:

  • Dipartimento di Scienze Sociali, Cognitive e Quantitative, Università di Modena e Reggio Emilia, via Giglioli Valle 9, 42100 Reggio Emilia, Italy;Dipartimento di Scienze Agrarie, della Formazione Primaria e della Programmazione Sanitaria, Università di Modena e Reggio Emilia, via Kennedy 17, 42100 Reggio Emilia, Italy

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2003

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Abstract

We investigate collineation groups of a finite projective plane of odd order n fixing an oval and having two orbits on it, one of which is assumed to be primitive. The situation in which the group fixes a point off the oval is considered. We prove that it occurs in a Desarguesian plane if and only if (n + 1)/2 is an odd prime, the group lying in the normalizer of a Singer cycle of PGL(2, n) in this case. For an arbitrary plane we show that the group cannot contain Baer involutions and derive a number of structural and numerical properties in the case where the group has even order. The existence question for a non-Desarguesian example is addressed but remains unanswered, although such an example cannot have order n ≤ 23 as computer searches carried out with GAP show.