ABSTRACT
A ‘subgeometry partition of a …nite projective space’ is simply a partition of the projective space by subgeometries isomorphic to …nite projective spaces of various dimensions over various …elds. The …rst examples of subgeometry partitions were partitions of
PG(2; q2) by subgeometries isomorphic to PG(2; q)’s, the so-called Baer subgeometry partitions. These partitions originated in the text [73] by Hirschfeld and Thas. There is a so-called lifting process, which we term ‘geometric lifting’ that produces from a Baer subgeometry partition a planar-spread that corresponds to a translation plane of order q3 with projective 2-spread in PG(5; q). The associated translation planes of order q3 and kernel containing GF (q) will often admit a collineation group that is ‡ag-transitive. Regarding the group action on the line at in…nity, a ‡ag-transitive group is simply a group that is transitive on the line at in…nity. In this case, and in the cases of most interest, this collineation group is cyclic of order q3 + 1. The subgroup of order q + 1 together with the kernel homology group of order q 1 together with the zero mapping generate a …eld isomorphic to GF (q), which contains a cyclic subgroup of order q2 1 that acts on the translation plane with orbits of length q+ 1. The existence of a group of order q2 1 containing the kernel group of order q 1, such that by adjoining the zero map one obtains a …eld R, over which the point set as a vector space is important to the theory. This retraction group is the key to the understanding of how to construct subgeometry partitions (see Johnson [113]). Basically, the orbits of length q + 1 ‘retract’ to subgeometries isomorphic to PG(2; q) that cover the associated projective space (the 6-dimensional vector space over GF (q) becomes 3-dimensional over R and produces the subgeometries isomorphic to PG(2; q2)). The geometric lifting process generalizes to partitions of PG(3; q2) by PG(1; q2)’s and PG(3; q)’s, the so-called mixed partitions and, in this case, produces translation planes of order q4
with kernel containing GF (q) and again there is an associated group of order q2 1 containing the kernel homology group of order q 1, and there is an associated …eld R, whose multiplicative group now has
component orbits of length 1 or q+1. The orbits of length q+1 retract to subgeometries isomorphic to PG(3; q), and the orbits of length 1 retract to subgeometries isomorphic to PG(1; q2). The existence of the retraction group with …eld isomorphic to GF (q2) was shown in Johnson [113]. Furthermore, in Jha and Johnson [84], it is shown that it is pos-
sible to use ‘algebraic lifting’to produce from a translation plane of order q2 with spread in PG(3; q), a corresponding spread of order q4 in PG(3; q2), wherein lies a retraction group. Considered over the subkernel isomorphic to GF (q), it turns out that there is a retraction group of order q2 1 with associated …eld isomorphic to GF (q2) that constructs an associated subgeometry partition of PG(3; q2) by PG(3; q)’s and PG(1; q2)’s: We shall use a variation of this process to construct new subgeometry partitions from focal-spreads.
