fundamental theorem of projective geometry
Theorem 1 (Fundamental Theorem of Projective Geometry I).
Every bijective order-preserving map (projectivity^{}) $f\mathrm{:}P\mathit{}G\mathit{}\mathrm{(}V\mathrm{)}\mathrm{\to}P\mathit{}G\mathit{}\mathrm{(}W\mathrm{)}$, where $V$ and $W$ are vector spaces^{} of finite dimension^{} not equal to 2, is induced by a semilinear transformation $\widehat{f}\mathrm{:}V\mathrm{\to}W$.
(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.
$P\mathrm{\Gamma}L(V)$ is the automorphism group^{} of the projective geometry^{}, $P\mathit{}G\mathit{}\mathrm{(}V\mathrm{)}$, of $V$, when $\mathrm{dim}\mathit{}V\mathrm{>}\mathrm{2}$.
Remark 3.
$P\mathrm{\Gamma}L(V)$ is the group of invertible^{} semi-linear transformations of a vector space $V$. (See classical groups (http://planetmath.org/Isometry2) for a full description of $P\mathit{}\mathrm{\Gamma}\mathit{}L\mathit{}\mathrm{(}V\mathrm{)}$.)
Notice that $\mathrm{Aut}PG(0,k)=1$ and $\mathrm{Aut}PG(1,k)=Sym(k\cup \{\mathrm{\infty}\})$. ($Sym(X)$ is the symmetric group^{} on the set $X$. $\mathrm{\infty}$ 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 $k=\mathbb{R}$ this corresponds to the vertical line through the origin, and so it has slope $\mathrm{\infty}$, 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 $dimV=2$ then $PG(V)$ 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 $PG(V)$ are order-preserving. Not all permutations arrise as semilinear maps however.
Example. If $k={\mathbb{Z}}_{p}$, 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 $V={k}^{2}$ so $\mathrm{Aut}PG(V)$ is the symmetric group on $p+1$ points, ${S}_{p+1}$. Yet the permutation $\pi $ mapping
$$\pi (1,0)=(0,1),\pi (0,1)=(1,0),\pi (x,y)=(x,y),(x,y)\notin \{(1,0),(0,1)\}$$ |
is therefore order-preserving by clearly non-linear, unless $p=2$. $\mathrm{\square}$
Remark 4.
There is a second form the fundamental theorem of projective geometry which appeals to the axiomatic construction of projective geometry.
References
- 1 Gruenberg, K. W. and Weir, A.J. Linear Geometry^{} 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 | Perspectivity^{} |