Gloria Rinaldi

On sharply vertex transitive 2-factorizations of the complete graph

We introduce the concept of a 2-starter in a group G of odd order. We prove that any 2-factorization of the complete graph admitting G as a sharply vertex transitive automorphism group is equivalent to a suitable 2-starter in G. Some classes of 2-starters are studied, with special attention given to those leading to solutions of some Oberwolfach or Hamilton-Waterloo problems.

Quaternionic Starters

Let m be an integer, m ≥ 2 and set n = 2m. Let G be a non-cyclic group of order 2n admitting a cyclic subgroup of order n. We prove that G always admits a starter and so there exists a one–factorization of K2n admitting G as an automorphism group acting sharply transitively on vertices. For an arbitrary even n > 2 we also show the existence of a starter in the dicyclic group of order 2n.

Hyperbolic unitals in the Hall planes

A new transformation method for incidence structures was introduced in [8],an open problem is to characterize classical incidence structures obtained by transformation of others. In this work we give some, sufficient conditions to transform, with the procedure of [8],a unital embedded in a projective plane into another one. As application of this result we construct unitals in the Hall planes by transformation of the hermitian curves and we give necessary and sufficient conditions for the constructed unitals to be projectively equivalent. This allows to find different classes of not projectively equivalent Buekenhouts unitals, [2],and to find the class of unitals descovered by Grüning, [4],easily proving its embeddability in the dual of a Hall plane. Finally we prove that the affine unital associated to the unital of [4]is isomorphic to the affine hyperbolic hermitian curve.

Automorphisms of (B)-geometries

Abstract(B)-Geometries are incidence structures arising from permutation sets. The present paper studies the automorphism groups of (B)-Geometries. In certain cases these automorphisms yield examples of inversive planes and of subplanes which are embedded in Minkowski planes (chapter 2). In chapter 3 we describe the automorphism groups of the (B)-Geometries arising from the groups PΓL(2, pn) and AΓL(1, pn) in their natural representations on the points of the projective and affine line.

A collection of results on Hamiltonian cycle systems with a nice automorphism group

Abstract We collect some old and new results on Hamiltonian cycle systems of the complete graph (or the complete graph minus a 1-factor) having an automorphism group that satisfies specific properties.

Construction of unitals in the Hall planes

Using a transformation method for incidence structures introduced in [9] I construct unitals embedded in the Hall planes by transformation of the Buekenhout-Metz unitals.

Minkowski near— planes

Finite Minkowski near-planes are defined and investigated. After giving a description of all the Minkowski near-planes of order 3 and 4 respectively, we will prove that each Minkowski near-plane of order n ≥ 5 is either a Minkowski plane of order n — 1 or is embeddable in a Minkowski plane of order n.

Arcs in Minkowski planes

The aim of this paper is to study some properties of k-arcs in Minkowski planes focalizing the attention on problems of existence and completness.

Primitive collineation groups of ovals with a fixed point

We investigate collineation groups of a finite projective plane of odd order n fixing an oval and having two orbits on it, one of which is assumed to be primitive. The situation in which the group fixes a point off the oval is considered. We prove that it occurs in a Desarguesian plane if and only if (n + 1)/2 is an odd prime, the group lying in the normalizer of a Singer cycle of PGL(2, n) in this case. For an arbitrary plane we show that the group cannot contain Baer involutions and derive a number of structural and numerical properties in the case where the group has even order. The existence question for a non-Desarguesian example is addressed but remains unanswered, although such an example cannot have order n ≤ 23 as computer searches carried out with GAP show.

Finite Minkowski planes and embedded inversive planes

Abstract. We show that each known finite Minkowski plane of order