ABSTRACT
As an important application of the theory of focal-spreads, we shall be constructing new subgeometry portions. As mentioned in the introductory remarks, the most important and relevant construction device of subgeometry partitions from vector spaces uses an appropriate …xedpoint-free group of order qd1, containing a scalar group of order q1, which is a …eld when adjoining the zero mapping. That is, from such a group acting on a partition of a vector space, a ‘retraction’ procedure produces what is called a quasi-subgeometry partition by Johnson [106] and a subgeometry partition when d = 2. The quasi-subgeometries are subgeometries if and only if each ‘line’is the full intersection with the set of ‘points’of a line of the projective superspace. This is the beauty of using groups …rst constructing quasi-subgeometries providing the bulk of the e¤ort with the only question remaining being whether we actually obtain subgeometries in the partition.
