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Projectivity

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Euclid

Posts : 95
Join date : 2009-09-15
Age : 56
Location : Alexandria

 Subject: Projectivity    Tue Sep 20, 2011 6:21 am How to generalize the Fano" = Klein equation ?there is a transitivity hierarchy (stick-n-stuck-all structures)1....... groups, affine spaces with translations1+ ... affine spaces with translations and dilatations2...... lines, fields3...... projective lines3+.... 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 pointsand 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/CollineationThe nice case is p prime, whenProjectiveLine(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 example9T15 - the affine line on 9 points, sharply 2-transitive (order 72)and6T19 - the affine line with twists (order 144)

Euclid

Posts : 95
Join date : 2009-09-15
Age : 56
Location : Alexandria

 Subject: Re: Projectivity    Tue Sep 20, 2011 6:51 am 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 byProjectivePlane" = 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

 Projectivity
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