ABSTRACT
In the previous chapters, we have given a variety of constructions of the so-called ‘transitive de…ciency one partial parallelisms, which are partial parallelisms in PG(3; q) of q2 + q spreads that admit a transitive group of the set of spreads. There is a unique extension to a parallelism, so the transitive group will act as collineation group of the extended plane (the ‘socle’plane), perhaps not faithfully. An understanding of these parallelisms is of considerable importance and interest and is also a wonderful blend of geometric, combinatorial, and group theoretic ideas. If q = pr, the transitive group G admits a Sylow p-group of order divisible by q. However, this group will be forced to ‘grow’to order q2 to accommodate the action on the spreads. When this occurs, it will turn out that every non-socle plane will be a translation plane with spread in PG(3; q) admitting a Baer group. Hence, we may apply the Johnson, Payne-Thas theorem that states the the plane is a derived conical ‡ock plane. This partial classi…cation theorem is due to Biliotti, Jha, and Johnson [19] who proved the result under a certain restriction on the Sylow p-subgroups (that they are ‘linear’) and to Diaz, Johnson, and Montinaro [45], who removed this restriction. The proof given here combines ideas of both of these papers. The background material required for reading the following theorem is listed in the appendix Chapter 41.
