ABSTRACT
In Chapter 10, a strong connection was shown between partial Desarguesian t-parallelisms over GF (q) and translation nets of order qzt covered by rational Desarguesian nets of degree 1 + qt that share a GF (q)-regulus net. The known Desarguesian 2-parallelisms are in PG(3; q) and admit a cyclic group C of order 1 + q + q2 acting transitively on the set of 1 + q + q2 2-spreads of the parallelism, and where q 2 mod 3, so that ( (q31)
(q1) ; q 1) = (q 1; 3) = 1. Also, there are transitive 2-parallelisms in PG(2r 1; 2), which we will present in a subsequent chapter. So all transitive 2-parallelisms are in PG(2r1; q), and in these cases ( (q
2r11) q1 ; q 1) = (2r 1; q 1) = 1. The following
was proved by Jha and Johnson in [93], [94], without the assumption (2r 1; q 1) = 1 (and without the assumption in the version of the theorem listed below) so there is potentially a gap in the proof given there. We shall give a somewhat brief version of the proof, using results of Lüneburg, Hering, and Ostrom.
