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 less than or equal to 23 as computer searches carried out with GAP show.

Primitive collineation groups of ovals with a fixed point / Bonisoli, Arrigo; Rinaldi, Gloria. - In: EUROPEAN JOURNAL OF COMBINATORICS. - ISSN 0195-6698. - STAMPA. - 24:7(2003), pp. 797-807. [10.1016/S0195-6698(03)00090-8]

Primitive collineation groups of ovals with a fixed point

BONISOLI, Arrigo;RINALDI, Gloria
2003

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 less than or equal to 23 as computer searches carried out with GAP show.
2003
24
7
797
807
Primitive collineation groups of ovals with a fixed point / Bonisoli, Arrigo; Rinaldi, Gloria. - In: EUROPEAN JOURNAL OF COMBINATORICS. - ISSN 0195-6698. - STAMPA. - 24:7(2003), pp. 797-807. [10.1016/S0195-6698(03)00090-8]
Bonisoli, Arrigo; Rinaldi, Gloria
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11380/303953
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