Projectivity

How to generalize the Fano" = Klein equation ?

there is a transitivity hierarchy (stick-n-stuck-all structures)

1....... groups, affine spaces with translations
1+ ... affine spaces with translations and dilatations
2...... lines, fields
3...... projective lines
3+.... also projective lines
4,5.... the sporadic Mathiew groups.

After groups and fields, the next symmetric objects are the projective lines.

Reading from the Conway-Hulpke-Mckay table :

9T27 - sharply triple transitive, the natural projective line with 9 points
and
9T32 - triple transitive, but also a projective group PgamaL(2,.


The extra-transitivity (more than three) of a projective line is explained by some twists of the line. Let' s say that the line is coordinated with some elements of a field; then, the automorphisms of that field will permute also the points on the line.

http://en.wikipedia.org/wiki/Collineation


The nice case is p prime, when
ProjectiveLine(p prime)'''= Lin (stick three stuck all)

for p^k +1 points on the projective line,
ProjectiveLine(p^k)''' = Cyc[k](Lin). (stick three and reach the twisting)

that Gal(K/k) is cyclic of order k

An affine line should be also twisted by field automorphisms,
for example
9T15 - the affine line on 9 points, sharply 2-transitive (order 72)and
6T19 - the affine line with twists (order 144)

The lines of a projective plane should be projective lines (at least this is the intention of every author).

Then a nice projective plane, for p prime, having p^2+p+1 points, is described by

ProjectivePlane" = Cyc[p-1] . AffinePlane

after pinning two points in a projective plane, a line is sketched. The remaining relabeling liberty comes from
- the rest of the projective line, and secondly from
- the remaining affine plane (without rotations) - after sticking one line, that could be considered the "infinite line", and thus blocking the directions