## Innovations in Incidence Geometry

### Groups of hyperovals in Desarguesian planes

#### Abstract

We show that if a hyperoval $ℋ$ of $PG ( 2 , q )$, $q > 4$, admits an insoluble group $G$, then $G$ fixes a subplane $π 0$ of order $q 0 > 2$, $ℋ$ meets $π 0$ in a regular hyperoval of $π 0$ on which $G ∩ PGL ( 3 , q )$ induces $PGL ( 2 , q 0 )$, and if $ℋ$ is not regular then $q > q 0 2$. We also bound above the order of the homography stabilizer of a non-translation hyperoval of $PG ( 2 , q )$ by $3 ( q − 1 )$. Finally, we show that the homography stabilizer of the Cherowitzo hyperovals is trivial for $q > 8$.

#### Article information

Source
Innov. Incidence Geom., Volume 6-7, Number 1 (2007), 37-51.

Dates
Accepted: 17 March 2008
First available in Project Euclid: 28 February 2019

https://projecteuclid.org/euclid.iig/1551323169

Digital Object Identifier
doi:10.2140/iig.2008.6.37

Mathematical Reviews number (MathSciNet)
MR2515257

Zentralblatt MATH identifier
1178.51006

Keywords
hyperoval group

#### Citation

Bayens, Luke; Cherowitzo, William; Penttila, Tim. Groups of hyperovals in Desarguesian planes. Innov. Incidence Geom. 6-7 (2007), no. 1, 37--51. doi:10.2140/iig.2008.6.37. https://projecteuclid.org/euclid.iig/1551323169

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