Irreducible collineation groups fixing a hyperoval
An automorphism group \(G\) of a finite projective plane \(\pi\) is called irreducible if \(G\) fixes no point, line or triangle of \(\pi\). The authors investigate the case that \(G\) fixes a hyperoval and that the order \(n\) of \(\pi\), which is necessarily even, is divisible by 4. The only known examples seem to be a conic plus its nucleus in \(PG(2,4)\) and the Lunelli-Sce-Hall hyperoval in \(PG(2,16)\). It turns out that the general situation closely resembles these two cases. If \(G\) contains involutory elations and \(n \not= 4\) then the group generated by all involutory elations modulo its center is isomorphic to its counterpart associated with the Lunelli-Sce-Hall hyperoval, i.e. it is a generalized Hessian group of order 18. This result indicates that the Lunelli-Sce-Hall hyperoval might be only the first representative of an infinite family of hyperovals and that this family can be constructed by an inductive process.
scientific article; zbMATH DE number 1836716
scientific article