fundamental theorem of projective geometry (original) (raw)

Theorem 1 (Fundamental Theorem of Projective Geometry I).

(Refer to [1, Theorem 3.5.5,Theorem 3.5.6].)

As an immediate corollary we notice that in fact V and W are vector spaces of the same dimension and over the isomorphic fields (or division rings). The dimension aspect is easily seen in other ways, and if the fields are finite fields so too is the entire corollary. However the true corollary to this theorem is

Corollary 2.

Notice that Aut⁡P⁢G⁢(0,k)=1 and Aut⁡P⁢G⁢(1,k)=S⁢y⁢m⁢(k∪{∞}). (S⁢y⁢m⁢(X) is the symmetric group on the set X. ∞ simply denotes a formal element outside of the field k which in many concrete instances does capture a conceptual notion of infinity. For example, when k=ℝ this corresponds to the vertical line through the origin, and so it has slope ∞, while the other elements of k are the slopes of the other lines.)

The Fundamental Theorem of Projective Geometry is in many ways “best possible.” For if dim⁡V=2 then P⁢G⁢(V) has only the two trivial subspace 0 and V – which cannot be moved by order preserving maps – and subspaces of dimension 1. Thus any two proper subspaces can be interchanged, transposed. So in this case all permutation of points in the projective line P⁢G⁢(V) are order-preserving. Not all permutations arrise as semilinear maps however.

Example.If k=ℤp, then there are no field automorphisms as k is a prime field. Hence all semilinear transforms are simply linear transforms. There are p+1 subspaces of dimension 1 in V=k2 so Aut⁡P⁢G⁢(V) is the symmetric group on p+1 points, Sp+1. Yet the permutation π mapping

is therefore order-preserving by clearly non-linear, unless p=2.□

References