ABSTRACT
In the construction of the parallelisms of Johnson (see Chapter 28), for K a …eld that admits a quadratic extension, there are two Pappian spreads and 0 in PG(3; K) that share a regulus R and the full central collineation group G with …xed axis ` of R of one of these spreads . If 0 denotes the associated Hall spread obtained by the derivation of R, then the parallelism is [ f0g; g 2 Gg. Hence, the parallelism is essentially determined by the two spreads ; and 0; and the group G. In a sense that we shall make clear below, the group G is ‘parallelism-inducing.’ The technique that we shall be discussing is suitable for when the base planes are Pappian, whereas, eventually we shall consider a generalization of the parallelism-inducing technique. When we consider ‘elation-switching,’ the generalization shall be essentially complete. Basically, the idea of the construction arises from the consideration of the proof that there is a cover of Baer subplanes of disjoint from the axis of a central collineation group by images of another Pappian spread under the central group G. This group acts sharply doubly transitively on the components of faxisg and acts sharply transitively on the Baer subplanes of incident with the zero vector and which are disjoint from the axis of G.
