Abstract. We establish the connections between finite projective planes admitting a collineation group of Lenz-Barlotti type 1.3 or 1.4, partially transitive planes of type (3) in the sense of Hughes, and planes admitting a quasiregular collineation group of type (g) in the DembowskiPiper classification; our main tool is an equivalent description by a certain type of difference set relative to disjoint subgroups which we will call a neo-difference set. We then discuss geometric properties and restrictions for the existence of planes of Lenz-Barlotti class 1.4. As a side result, we also obtain a new synthetic description of projective triangles in desarguesian planes.
Abstract.A tube of even order q = 2 a is a set T = {L, £} ofq + 3 pairwise skew lines in PG(3, q) such that every plane on L meets the lines of £ in a hyperoval. The quadric tube is obtained as follows. Take a hyperbolic quadric Q = Q+(q) in PG(3, q); let L be an exterior line, and let £ consist of the polar line of L together with a regulus on Q.In this paper we show the existence of tubes of even order other than the quadric one, and we prove that the subgroup of PFL(4, q) fixing a tube {L, /2} cannot act transitively on/2. As pointed out by a construction due to Pasini, this implies new results for the existence of flat ~r.Cz geometries whose C2-residues are nonclassical generalized quadrangles different from nets. We also give the results of some computations on the existence and uniqueness of tubes in PG(3, q) for small q. Further, we define tubes for odd q (replacing 'hyperoval' by 'conic' in the definition), and consider briefly a related extremal problem.Mathematics Subject Classifications (1991): Primary: 51E20; Secondary 51 E24, 05B25.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.