fundamental theorem of projective geometry

Theorem 1 (Fundamental Theorem of Projective Geometry I).

Every bijective order-preserving map (projectivityMathworldPlanetmath) f:PG(V)PG(W), where V and W are vector spacesMathworldPlanetmath of finite dimensionPlanetmathPlanetmath not equal to 2, is induced by a semilinear transformation f^:VW.

(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 isomorphicPlanetmathPlanetmathPlanetmathPlanetmath fields (or division rings). The dimension aspect is easily seen in other ways, and if the fields are finite fieldsMathworldPlanetmath so too is the entire corollary. However the true corollary to this theorem is

Corollary 2.

PΓL(V) is the automorphism groupMathworldPlanetmath of the projective geometryMathworldPlanetmath, PG(V), of V, when dimV>2.

Remark 3.

PΓL(V) is the group of invertiblePlanetmathPlanetmath semi-linear transformations of a vector space V. (See classical groups ( for a full description of PΓL(V).)

Notice that AutPG(0,k)=1 and AutPG(1,k)=Sym(k{}). (Sym(X) is the symmetric groupMathworldPlanetmathPlanetmath 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= 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 GeometryMathworldPlanetmath is in many ways “best possible.” For if dimV=2 then PG(V) has only the two trivial subspacePlanetmathPlanetmathPlanetmath 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 permutationMathworldPlanetmath of points in the projective line PG(V) are order-preserving. Not all permutations arrise as semilinear maps however.

Example. If k=p, then there are no field automorphisms as k is a prime fieldMathworldPlanetmath. Hence all semilinear transforms are simply linear transforms. There are p+1 subspaces of dimension 1 in V=k2 so AutPG(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.

Remark 4.

There is a second form the fundamental theorem of projective geometry which appeals to the axiomatic construction of projective geometry.


  • 1 Gruenberg, K. W. and Weir, A.J. Linear GeometryMathworldPlanetmath 2nd Ed. (English) [B] Graduate Texts in Mathematics. 49. New York - Heidelberg - Berlin: Springer-Verlag. X, 198 p. DM 29.10; $ 12.80 (1977).
  • 2 Kantor, W. M. Lectures notes on Classical Groups.
Title fundamental theorem of projective geometry
Canonical name FundamentalTheoremOfProjectiveGeometry
Date of creation 2013-03-22 15:51:14
Last modified on 2013-03-22 15:51:14
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 17
Author Algeboy (12884)
Entry type Theorem
Classification msc 51A10
Classification msc 51A05
Related topic PerspectivityMathworldPlanetmath