ABSTRACT
In this chapter, we use the theory that we have developed regarding Baer groups and their representation. We also connect algebraic lifting and geometric lifting, which are as disparate concepts as one could imagine: one process, that of lifting a spread in PG(3; q) de…ning a translation plane of order q2 to a corresponding spread in PG(3; q2) that provides a translation plane of order q4, and the second process, that of lifting a subgeometry partition to a translation plane. We also are able to use the algebraic lifting process in combination with our Sperner spread construction theorem presented in Chapter 4 for the construction of still more, interesting subgeometry partitions. There is a fusion process in the theory of semi…elds that allows a coordinate change that fuses certain sub…elds of the nuclei. Here we consider a similar fusion process for Baer groups, noting that a Baer group will arise from a group associated with a nucleus upon derivation (usually of a semi…eld plane). Using these ideas, there will be a ubiquity of subgeometry partitions that arise from the concept of double-Baer groups. In the previous chapters, we have tied together a number of various
spreads that admit retraction groups, thereby constructing subgeometry partitions. The natural question is how general are the spreads that give rise to subgeometry partitions. Or, we may ask the opposite question: given a t-spread, is there a subgeometry partition that gives rise to it?
