Collineation groups strongly irreducible on an oval in a projective plane of odd order

Strongly irreducibile collineation groups have played an important role in the study of collineation groups of a finite projective plane. The theory of strongly irreducible collineation groups containing perspectivities is well developed, but not many results have been obtained so far when the hypothesis on the existence of perspectivities is dropped. It is also of interest to investigate "local" versions of irreducibility. In this regard, in a finite projective plane pi containing an oval Omega, a collineation group G is strongly irreducible on 12 if G preserves Omega but it does not leave invariant any point, chord or suboval of Omega. In even order projective planes, collineation groups strongly irreducible on an oval were thoroughly investigated by Biliotti and Korchmaros. In this paper we investigate strongly irreducible collineation groups G on an oval SI in a projective plane pi of order n with n 1 (mod 4). Let H be the subgroup of G consisting of all collineations inducing an even permutation on Omega. We prove that the possibilities for H are essentially two: either H contains a normal subgroup isomorphic to PSL(2, q) with q >= 5 odd, or H = O(G) >= S-2 where S-2 is a Sylow 2-subgroup of G.