Intransitive collineation groups of ovals fixing a triangle

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 p be a finite projective plane of odd order n containing an oval O: If a collineation group G of p satisfies the properties: (a) G fixes O and the action of G on O yields precisely two orbits O1 and O2; (b) G has even order and a faithful primitive action on O2; (c) G fixes neither points nor lines but fixes a triangle ABC in which the points A; B; C are not on the oval O; then nAf7; 9; 27g; the orbit O2 has length 4 and G acts naturally on O2 as A4 or S4: Each order nAf7; 9; 27g 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 1⁄4 7; 9; the determination of the planes is still incomplete for n 1⁄4 27: r 2003 Elsevier Science (USA). All rights reserved.