2 . Mauro Biliotti and Alessandro Montinaro Universitá degli Studi - Lecce mauro

An interesting question posed by Cofman [Atti Accad. Naz. Lincei, (8) 43 (1967)] is the following: ”Let Π be a finite plane with a subset O of points admitting a collineation group G which maps O onto itself and induces a doubly transitive collineation group on the points of O. What can we say about the plane, the set O and the collineation group G?”. A survey about the knowledge on this subject at that time was given by Cofman in the same paper. Clearly, either the structure of a non-trivial 2-design is induced on O, or O is an arc or O is contained in a line. A fundamental contribution to the problem is certainly the Kantor’s classification of 2-designs with an automorphism group 2-transitive on the points [J. Comb. Theory Ser. A 38 (1985)]. After the Cofman’s paper, several conclusive results have been obtained by Korchmaros and Kantor when O is an oval or a unital. Relevant results have also been achieved when O consists of a line or a line minus a point by Korchmaros and Hiramine. Recently, Ganley, Jha and Johnson have essentially classified the line-sized sets O in a translation plane for G nonsolvable. The case when O is the line at infinity of a translation plane has been completely settled. In a forthcoming paper Biliotti and Francot [J. Geom.], the authors attempt to classify, under the Cofman’s assumptions, the triples (Π,O, G), where the G-orbit O has length at least the order n of Π. In that paper a survey of the known results is also given. When the length of O is smaller than n, but rather close to n, it still seems possible to obtain characterization results. A first result in this way is contained in a forthcoming paper of Biliotti and Montinaro [Adv. Geom.] in which a complete classification for (Π,O, G) is given when O has length n− 3. In the present talk, we expose the case where the length of O is at least n 2 . While the case when O is an arc is in progress, the cases when O is a 2-designs or it is contained in a line are quite complete and there are several examples.