ABSTRACT
In Chapter 3, we developed the notion of an extended André spread, and we noted there were normally nice groups attached with such partitions. Since subgeometry partitions arise directly due to certain …xedpoint-free groups, it certainly is possible to construct subgeometry partitions from extended André spreads. There is really an embarrassing wealth of wonderful subgeometry partitions that can be generated and we sketch only a few of these in this text. Let be a Desarguesian r-spread of order qsn. We may regard as
an rsn-vector space over GF (q). Moreover, since the kernel homology group Ks determines a …eld Ks [ f0g = K+s , we may also regard as an rn-vector space over K+s : Letting
d : (x1; x2; ::; xr) 7! (x1d; x2d; ::; xrd); d 2 GF (qs), with 0, the zero mapping,
then K+s = h d; d 2 GF (qs)i. De…ne dv = d v = (x1d; x2d; ::; xrd), for v = (x1; x2; ::; xr); then clearly is an rn-dimensional K+s -space. Let the lattice of subspaces be denoted by PG(rn 1; qs). Then is an rn-dimensional subspace over K+s and the lattice of vector subspaces forms a projective space PG(rn 1; qs). The Main Theorem on Extended André Subgeometries is as follows:
Theorem 29. Let the lattice of vector subspaces of over K+s be denoted by PG(rn 1; qs). Take any
= n y = (xq
1 n1d 1q1 ; ::; xq
rj1 1 nrj1d
1qrj1 ); d 2 GF (qsn) o :
Again, there are (qsn 1)=(q(1;::;rj1) 1) = (qs 1)=(q(1;2;::;rj1;s) 1)
component orbits of length
(qs 1)=(q(1;2;::;rj1;s) 1) under Ks.
