Groups of hyperovals in Desarguesian planes
type: Item
instance of: scholarly article
title: Groups of hyperovals in Desarguesian planes
zbMATH Open document ID: 1178.51006
author: Luke Bayens
author: William E. Cherowitzo
author: Tim Penttila
published in: Innovations in Incidence Geometry
publication date: 2009-06-23T00:00:00Z
review text: The authors prove a number of results about the automorphism groups of hyperovals in Desarguesian planes, paying special attention to hyperovals admitting insoluble groups of automorphisms. The main result shows that if \({\mathcal H}\) is a hyperoval with an insoluble group of automorphisms \(G \subseteq P\Gamma L(3,q)\), \(q \geq 4\), then \(G\) fixes a subplane \(\pi_0\) of order \(q_0\), \(\pi_0 \cap {\mathcal H}\) is a regular hyperoval of \(\pi_0\), and \(G\) has a normal subgroup isomorphic to \(\text{PGL}(2,q_0)\). They further show that a hyperoval of \(\text{PG}(2,q)\) with homography stabilizer of order greater than \(3(q-1)\) is a translation hyperoval. Finally, they use computer software to verify that the homography group of a Cherowitzo hyperoval is trivial for \(q>8\).
Mathematics Subject Classification ID: 51E20
Mathematics Subject Classification ID: 51E21
zbMATH DE Number: 5567853
zbMATH Keywords: hyperoval
zbMATH Keywords: automorphism group
zbMATH Keywords: Cherowitzo hyperoval
reviewed by: Alberto Luis Delgado
MaRDI profile type: Publication
full work available at URL: iig.2008.6.37
OpenAlex ID: W2915294095
Wikidata QID: Q128348044
DOI: 10.2140/IIG.2008.6.37
description: scientific article; zbMATH DE number 5567853