Noname manuscript No. (will be inserted by the editor) The group fixing a completely regular line–oval Antonio Maschietti the date of receipt and acceptance should be inserted later Keywords line–oval, translation plane, symplectic spread Mathematics Subject Classification (2000) Primary 51A35, 51A40, 51E20; Secondary 51A50 Abstract We prove that the action of the full collineation group of a symplectic translation plane of even order on the set of completely regular line–ovals is transitive. This provides us with a complete description of the group of collineations fixing a completely regular line–oval. 1 Introduction Symplectic translation planes of even order are interesting geometrical and combina- torial objects, because of their close relation with non–linear codes ([1], [2], [3], [5]). Recently we gave a necessary and sufficient condition for a finite translation plane of even order to be symplectic in terms of the existence of completely regular line–ovals (see [10], [11] and the next section). A line–conic in a desarguesian affine plane is the basic, and unique, example of a completely regular line–oval (see [9]). Line–conics are equivalent to one another with respect to the full collineation group of the plane. There- fore it is natural to ask whether such a result holds for completely regular line–ovals in any symplectic translation plane. In this paper we investigate the action of the full collineation group of the symplectic translation plane on the set of all completely regular line–ovals. The main result is that such an action is transitive. Also, this result provides us with a complete description of the group of all collineations fixing a completely regular line–oval: it is isomorphic to the group fixing the symplectic spread, modulo K ∗ , the kernel homology group. As a consequence we prove that the number of completely regular line–ovals is |T ||K ∗ |, where T is the translation group of the plane. Antonio Maschietti Dipartimento di Matematica “G. Castelnuovo” La Sapienza Universita’ di Roma p.le A. Moro 5, 00185 Roma, Italy E-mail: maschiet@mat.uniroma1.it