Parallelism-inducing groups (Q1396534)

Let \(K\) be a field. A parallelism of \(\text{ PG}(3,K)\) is a set of spreads partitioning the line set. If \(\Sigma\) and \(\Sigma'\) are two distinct spreads in \(\text{ PG}(3,K)\) that share exactly a regulus \(R\), then a collineation group \(G\) of the affine plane associated with \(\Sigma\) that leaves a line \(l\) of \(R\) invariant is said to be a parallellism-inducing group with respect to \(\Sigma\) and \(\Sigma'\) if it satisfies the following properties: (i) \(G\) is sharply 2-transitive on the set of components of \(\Sigma\) distinct from \(l\), (ii) \(G\) is regular on the set of Baer subplanes of the affine plane associated with \(\Sigma\) which are disjoint from \(l\), (iii) \(G_R\) fixes \(\Sigma'\) and acts regularly on the components of \(\Sigma' - R\). A Pappian spread in \(\text{ PG}(3,K)\) is a spread whose corresponding affine plane may be coordinatized by a field that is a quadratic extension of \(K\). In the paper, the authors use parallellism-inducing groups of Pappian spreads to construct a variety of new parallelisms in \(\text{ PG}(3,K)\).