Intransitive collineation groups of ovals fixing a triangle

Abstract

We investigate collineation groups of a finite projective plane of odd order fixing an oval and having two orbits on it, one of which is assumed to be primitive. The situation in which there exists a fixed triangle off the oval is considered in detail. Our main result is the following. Theorem. Let π be a finite projective plane of odd order n containing an oval Ω. If a collineation group G of π satisfies the properties: (a) G fixes Ω and the action of G on Ω yields precisely two orbits Ω 1 and Ω 2 , (b) G has even order and a faithful primitive action on Ω 2 , (c) G fixes neither points nor lines but fixes a triangle ABC in which the points A, B, C are not on the oval Ω, then n∈{7,9,27}, the orbit Ω 2 has length 4 and G acts naturally on Ω 2 as A 4 or S 4. Each order n∈{7,9,27} does furnish at least one example for the above situation; the determination of the planes and the groups which do occur is complete for n=7, 9; the determination of the planes is still incomplete for n=27.