ABSTRACT
In Chapter 63, it has been pointed out that the Fisher planes, corresponding to the Fisher flocks of a quadratic cone in PG(3, q) correspond to the q-nest planes. The Fisher planes admit a collineation group of order q(q + 1) and indeed, any translation plane of order q2 with spread in PG(3, q) that admits an affine elation group of order q is potentially a conical flock plane, and is one, provided the elation group has orbits defining reguli. However, in general a translation plane of order q2
admitting a collineation group of order q(q + 1) may not be Fisher, but there may be a more general connection with Desarguesian planes and construction methods by use of sets of reguli. The general connection involves what are called ‘multiple nests’, the existence of which is not completely clear, although certain of these structures do occur in a natural manner.
