fundamental theorem of projective geometry

fundamental theorem of projective geometry

Theorem 1

(Fundamental Theorem of Projective Geometry I).

Every bijective order-preserving map (

projectivity

)

f

:

P

⁢

G

⁢

(

V

)

→

P

⁢

G

⁢

(

W

)

, where

V

and

W

are

vector spaces

of finite

dimension

not equal to 2, is

induced

by a

semilinear transformation

f

^

:

V

→

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

⁢

Γ

⁢

L

⁢

(

V

)

is the

automorphism group

of the

projective geometry

,

P

⁢

G

⁢

(

V

)

, of

V

, when

dim

⁡

V

>

2

.

Remark 3

.

P

⁢

Γ

⁢

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

⁢

Γ

⁢

L

⁢

(

V

)

.)

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

=

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

Aut

⁡

P

⁢

G

⁢

(

V

)

is the symmetric group on

p

+

1

points,

S

p

+

1

. Yet the permutation

π

mapping

π

⁢

(

1

,

0

)

=

(

0

,

1

)

,

π

⁢

(

0

,

1

)

=

(

1

,

0

)

,

π

⁢

(

x

,

y

)

=

(

x

,

y

)

,

(

x

,

y

)

∉

{

(

1

,

0

)

,

(

0

,

1

)

}

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.

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

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