# fundamental theorem of projective geometry

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

###### Remark 3.

$P\Gamma L(V)$ 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 $P\Gamma L(V)$.)

Notice that $\operatorname{Aut}PG(0,k)=1$ and $\operatorname{Aut}PG(1,k)=Sym(k\cup\{\infty\})$. ($Sym(X)$ is the symmetric group   on the set $X$. $\infty$ 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 $\infty$, 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 $\dim V=2$ 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 $\operatorname{Aut}PG(V)$ is the symmetric group on $p+1$ points, $S_{p+1}$. Yet the permutation $\pi$ mapping

 $\pi(1,0)=(0,1),\quad\pi(0,1)=(1,0),\quad\pi(x,y)=(x,y),\qquad(x,y)\notin\{(1,0% ),(0,1)\}$

is therefore order-preserving by clearly non-linear, unless $p=2$. $\Box$

###### Remark 4.

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

## References

Title fundamental theorem of projective geometry FundamentalTheoremOfProjectiveGeometry 2013-03-22 15:51:14 2013-03-22 15:51:14 Algeboy (12884) Algeboy (12884) 17 Algeboy (12884) Theorem msc 51A10 msc 51A05 Perspectivity  