ABSTRACT
In this part, we give a variety of techniques for the construction of parallelisms, all based on what are called ‘parallelism-inducing groups.’ The basic idea is to try to construct parallelisms that arise by …nding a group that …xes a given spread and which acts transitively on the remaining spreads. The basic ideas work for both …nite and in…nite parallelisms. Once it is seen how to construct such ‘de…ciency one transitive partial parallelisms,’we see that the …xed spread is forced to be Desarguesian in the …nite case. The process constructing the Johnson parallelisms used two distinct Pappian spreads that share exactly a regulus in PG(3; K), and the group used is a central collineation group of one of these spreads thus permuting the remaining Pappian spreads to create a regulus. However, the spreads in the transitive orbit are not disjoint form the original Pappian spread but derivation of the reguli in the transitive group will make the corresponding Hall spreads disjoint on lines, thus creating a parallelism with one Pappian spread and the remaining spreads Hall. This can be generalized by taking a sequence of Pappian spreads sharing a regulus and using cosets of the group. This creates the idea of an m-parallelism. In the sequence of m Pappian spreads if there are n distinct spreads, this produces what is called an ‘(m;n)-parallelisms.’ The results of this part are modi…cations of results of the author and R. Pomareda [153],[154][152],[150].
