COLLINEATION GROUPS WHICH ARE PRIMITIVE ON AN OVAL OF A PROJECTIVE PLANE OF ODD ORDER MAURO BILIOTTI AND GABOR KORCHMAROS ABSTRACT It is shown that a projective plane of odd order, with a collineation group acting primitively on the points of an invariant oval, must be desarguesian. Moreover, the group is actually doubly transitive, with only one exception. The main tool in the proof is that a collineation group leaving invariant an oval in a projective plane of odd order has 2-rank at most three. Introduction Let n be a projective plane of odd order, Q an oval of n and G a collineation group of n leaving Q invariant. In 1967 Cofman [4] showed that if (1) G acts doubly transitively on Q, (2) all involutions in G are homologies, then n is desarguesian of order q, Q is a conic and G contains PSL (2, q). Afterwards, Kantor [15] proved that condition (2) can be weakened by assuming only that G contains involutorial homologies. Finally, in 1978 Korchmaros [17] showed that condition (2) is quite unnecessary, as already conjectured by Dembowski in his famous book [5]. Obviously the question arises as to how far condition (1) itself can be weakened to achieve a similar result. In this direction here we prove the following theorem. MAIN THEOREM. Let n be a finite projective plane of odd order n, let Cl be an oval of n and let G be a collineation group of n leaving Q invariant and acting primitively on its points. Then n is desarguesian, Q is a conic, and either n = q and G contains a normal subgroup acting on Q as PSL (2, q) in its usual doubly transitive representation, or n = 9 and G acts on Q as A 5 or S 5 in their primitive representation of degree 10. The latter possibility actually occurs in the desarguesian plane of order 9. The proof of the Main Theorem involves some properties about Sylow 2-subgroups of a collineation group leaving invariant an oval which may be of independent interest. They are collected in §2. 1 Throughout the paper n denotes a finite projective plane of odd order n. We refer to [5] for standard notions about projective planes. Here we only recall that, following Hering [10], a collineation group G of n is said to be irreducible over n if it does not fix any point, line or triangle and strongly irreducible over n if it is irreducible and does not fix any proper subplane. Received 29 March 1985; revised 26 November 1985. 1980 Mathematics Subject Classification 51 El5. J. London Math. Soc. (2) 33 (1986) 525-534