Peter J. Cameron

Queen Mary and West¬eld College

2000

Preface

It is common now in academic circles to lament the decline in the teaching of

geometry in our schools and universities, and the resulting loss of “geometric in-

tuition” among our students. On the other hand, recent decades have seen renewed

links between geometry and physics, to the bene¬t of both disciplines. One of the

world™s leading mathematicians has argued that the insights of “pre-calculus” ge-

ometry have a rˆ le to play at all levels of mathematical activity (Arnol™d [A]).

o

There is no doubt that a combination of the axiomatic and the descriptive ap-

proaches associated with algebra and geometry respectively can help avoid the

worst excesses of either approach alone.

These notes are about geometry, but by no means all or even most of geom-

etry. I am concerned with the geometry of incidence of points and lines, over an

arbitrary ¬eld, and unencumbered by metrics or continuity (or even betweenness).

The major themes are the projective and af¬ne spaces, and the polar spaces asso-

ciated with sesquilinear or quadratic forms on projective spaces. The treatment

of these themes blends the descriptive (What do these spaces look like?) with the

axiomatic (How do I recognize them?) My intention is to explain and describe,

rather than to give detailed argument for every claim. Some of the theorems (es-

pecially the characterisation theorems) are long and intricate. In such cases, I give

a proof in a special case (often over the ¬eld with two elements), and an outline

of the general argument.

The classical works on the subject are the books of Dieudonn´ [L] and Artin [B].

e

I do not intend to compete with these books. But much has happened since

they were written (the axiomatisation of polar spaces by Veldkamp and Tits (see

Tits [S]), the classi¬cation of the ¬nite simple groups with its many geometric

spin-offs, Buekenhout™s geometries associated with diagrams, etc.), and I have

included some material not found in the classical books.

Roughly speaking, the ¬rst ¬ve chapters are on projective spaces, the last

¬ve on polar spaces. In more detail: Chapter 1 introduces projective and af¬ne

i

ii

spaces synthetically, and derives some of their properties. Chapter 2, on projective

planes, discusses the rˆ le of Desargues™ and Pappus™ theorems in the coordinati-

o

sation of planes, and gives examples of non-Desarguesian planes. In Chapter 3,

we turn to the coordinatisation of higher-dimensional projective spaces, follow-

ing Veblen and Young. Chapter 4 contains miscellaneous topics: recognition of

some subsets of projective spaces, including conics over ¬nite ¬elds of odd char-

acteristic (Segre™s theorem); the structure of projective lines; and generation and

simplicity of the projective special linear groups. Chapter 5 outlines Buekenhout™s

approach to geometry via diagrams, and illustrates by interpreting the earlier char-

acterisation theorems in terms of diagrams.

Chapter 6 relates polarities of projective spaces to re¬‚exive sesquilinear forms,

and gives the classi¬cation of these forms. Chapter 7 de¬nes polar spaces, the

geometries associated with such forms, and gives a number of these properties;

the Veldkamp“Tits axiomatisation and the variant due to Buekenhout and Shult

are also discussed, and proved for hyperbolic quadrics and for quadrics over the

2-element ¬eld. Chapter 8 discusses two important low-dimensional phenomena,

the Klein quadric and triality, proceeding as far as to de¬ne the polarity de¬ning

the Suzuki“Tits ovoids and the generalised hexagon of type G2 . In Chapter 9, we

take a detour to look at the geometry of the Mathieu groups. This illustrates that

there are geometric objects satisfying axioms very similar to those for projective

and af¬ne spaces, and also having a high degree of symmetry. In the ¬nal chapter,

we de¬ne spinors and use them to investigate the geometry of dual polar spaces,

especially those of hyperbolic quadrics.

The notes are based on postgraduate lectures given at Queen Mary and West-

¬eld College in 1988 and 1991. I am grateful to members of the audience on these

occasions for their comments and especially for their questions, which forced me

to think things through more carefully than I might have done. Among many

pleasures of preparing these notes, I count two lectures by Jonathan Hall on his

beautiful proof of the characterisation of quadrics over the 2-element ¬eld, and

the challenge of producing the diagrams given the constraints of the typesetting

system!

In the introductory chapters to both types of spaces (Chapters 1 and 6), as well

as elsewhere in the text (especially Chapter 10), some linear algebra is assumed.

Often, it is necessary to do linear algebra over a non-commutative ¬eld; but the

differences from the commutative case are discussed. A good algebra textbook

(for example, Cohn (1974)) will contain what is necessary.

Peter J. Cameron, London, 1991

iii

Preface to the second edition

Materially, this edition is not very different from the ¬rst edition which was

published in the QMW Maths Notes series in 1991. I have converted the ¬les

into L TEX, corrected some errors, and added some new material and a few more

A

references; this version does not represent a complete bringing up-to-date of the

original. I intend to publish these notes on the Web.

In the meantime, one important relevant reference has appeared: Don Taylor™s

book The Geometry of the Classical Groups [R]. (Unfortunately, it has already

gone out of print!) You can also look at my own lecture notes on Classical Groups

(which can be read in conjunction with these notes, and which might be integrated

with them one day). Other sources of information include the Handbook of Inci-

dence Geometry [E] and (on the Web) two series of SOCRATES lecture notes at

http://dwispc8.vub.ac.be/Potenza/lectnotes.html

and

http://cage.rug.ac.be/˜fdc/intensivecourse2/final.html

Please note that, in Figure 2.3, there are a few lines missing: dotted lines utq

and urv and a solid line ub1 c2 . (The reason for this is hinted at in Exercise 3 in

Section 1.2.)

Peter J. Cameron, London, 2000

iv

Contents

1 Projective spaces 1

1.1 Fields and vector spaces . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Projective spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 The “Fundamental Theorem of Projective Geometry” . . . . . . . 8

1.4 Finite projective spaces . . . . . . . . . . . . . . . . . . . . . . . 14

2 Projective planes 19

2.1 Projective planes . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Desarguesian and Pappian planes . . . . . . . . . . . . . . . . . . 22

2.3 Projectivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Coordinatisation of projective spaces 31

3.1 The GF 2¡ case . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2 An application . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.5 Af¬ne spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.6 Transitivity of parallelism . . . . . . . . . . . . . . . . . . . . . . 43

4 Various topics 45

4.1 Spreads and translation planes . . . . . . . . . . . . . . . . . . . 45

4.2 Some subsets of projective spaces . . . . . . . . . . . . . . . . . 48

4.3 Segre™s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.4 Ovoids and inversive planes . . . . . . . . . . . . . . . . . . . . . 57

4.5 Projective lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.6 Generation and simplicity . . . . . . . . . . . . . . . . . . . . . . 62

v

CONTENTS

vi

5 Buekenhout geometries 65

5.1 Buekenhout geometries . . . . . . . . . . . . . . . . . . . . . . . 65

5.2 Some special diagrams . . . . . . . . . . . . . . . . . . . . . . . 70

6 Polar spaces 75

6.1 Dualities and polarities . . . . . . . . . . . . . . . . . . . . . . . 75

6.2 Hermitian and quadratic forms . . . . . . . . . . . . . . . . . . . 81

6.3 Classi¬cation of forms . . . . . . . . . . . . . . . . . . . . . . . 84

6.4 Classical polar spaces . . . . . . . . . . . . . . . . . . . . . . . . 88

6.5 Finite polar spaces . . . . . . . . . . . . . . . . . . . . . . . . . 92

7 Axioms for polar spaces 97

7.1 Generalised quadrangles . . . . . . . . . . . . . . . . . . . . . . 97

7.2 Diagrams for polar spaces . . . . . . . . . . . . . . . . . . . . . 101

7.3 Tits and Buekenhout“Shult . . . . . . . . . . . . . . . . . . . . . 105

7.4 Recognising hyperbolic quadrics . . . . . . . . . . . . . . . . . . 107

7.5 Recognising quadrics over GF 2¡ . . . . . . . . . . . . . . . . . . 109

7.6 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . 113

8 The Klein quadric and triality 115

8.1 The Pfaf¬an . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

8.2 The Klein correspondence . . . . . . . . . . . . . . . . . . . . . 117

8.3 Some dualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

8.4 Dualities of symplectic quadrangles . . . . . . . . . . . . . . . . 123

8.5 Reguli and spreads . . . . . . . . . . . . . . . . . . . . . . . . . 127

8.6 Triality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

8.7 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

8.8 Generalised polygons . . . . . . . . . . . . . . . . . . . . . . . . 131

8.9 Some generalised hexagons . . . . . . . . . . . . . . . . . . . . . 133

9 The geometry of the Mathieu groups 137

9.1 The Golay code . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

9.2 The Witt system . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

9.3 Sextets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

9.4 The large Mathieu groups . . . . . . . . . . . . . . . . . . . . . . 143

9.5 The small Mathieu groups . . . . . . . . . . . . . . . . . . . . . 144

CONTENTS vii

10 Exterior powers and Clifford algebras 147

10.1 Tensor and exterior products . . . . . . . . . . . . . . . . . . . . 147

10.2 The geometry of exterior powers . . . . . . . . . . . . . . . . . . 150

10.3 Near polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

10.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 155

10.4 Dual polar spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 155

10.5 Clifford algebras and spinors . . . . . . . . . . . . . . . . . . . . 156

10.6 The geometry of spinors . . . . . . . . . . . . . . . . . . . . . . 159

Index 168

1

Projective spaces

In this chapter, we describe projective and af¬ne spaces synthetically, in terms of

vector spaces, and derive some of their geometric properties.

1.1 Fields and vector spaces

Fields will not necessarily be commutative; in other words, the term “¬eld”

will mean “division ring” or “skew ¬eld”, while the word “commutative” will be

used where necessary. Often, though, I will say “skew ¬eld”, as a reminder. (Of

course, this refers to the multiplication only; addition will always be commuta-

tive.)

Given a ¬eld F, let

© ¨ ¢ §¦¥ ¤¢ ¡ ¢ ¡

± Fn± ©

I n : 0 n : n 1F 0

£ £

¨¥

Then I is an ideal in , hence I c for some non-negative integer c called the

£

characteristic of F. The characteristic is either 0 or a prime number. For each

value of the characteristic, there is a unique prime ¬eld which is a sub¬eld of any

¬eld of that characteristic: the rational numbers in characteristic zero, and the

integers modulo p in prime characteristic p.

Occasionally I will assume rudimentary results about ¬eld extensions, degree,

and so on.

Much of the time, we will be concerned with ¬nite ¬elds. The main results

about these are as follows.

Theorem 1.1 (Wedderburn™s Theorem) A ¬nite ¬eld is commutative.

1

2 1. Projective spaces

Theorem 1.2 (Galois™ Theorem) A ¬nite ¬eld has prime power order. For any

prime power q, there is a unique ¬nite ¬eld of order q.

The unique ¬eld of order q is denoted by GF¥ q¨ . If q pd with p prime,

its additive structure is that of a d-dimensional vector space over its prime ¬eld

GF¥ p¨ (the integers modulo p). Its multiplicative group is cyclic (of order q 1),

and its automorphism group is cyclic (of order d). If d 1 (that is, if q is prime),

then GF q is the ring of integers mod q.

¨¥

An anti-automorphism of a ¬eld is a bijection σ with the properties

σ

cσ cσ

c c2

¥ ¨

1

1 2

σ σσ

c c2 c2 c1

¨©¥ ©

1

The identity (or, indeed, any automorphism) is an anti-automorphism of a com-

mutative ¬eld. Some non-commutative ¬elds have anti-automorphisms. A well-

known example is the ¬eld of quaternions, with a basis over consisting of

elements 1 i j k satisfying

i2 j2 k2 1 ij k jk i ki j;

the anti-automorphism is given by

a bi cj dk a bi cj dk

Others, however, do not.

The opposite of the ¬eld F is the ¬eld F , where the binary oper-

!¨© ¥ ¨ "# ¥

ation is de¬ned by the rule

"

c1 c2 c2 c1 ©

"

Thus, an anti-automorphism of F is just an isomorphism between F and its oppo-

site F . $

For non-commutative ¬elds, we have to distinguish between left and right

vector spaces. In a left vector space, if we write the product of the scalar c and the

¥

vector v as cv, then c1 2 v¨

c 1 c2 v holds. In a right vector space, this condition

c ¥ ¨

¥

reads c1 2 v¨

c 2 c1 v. It is more natural to write the scalars on the right (thus:

c

¥ ¨

¨¥

vc), so that the condition is v c2 c1 v¥ c2 c1 ). A right vector space over F is a ¨

left vector space over F . $

Our vector spaces will almost always be ¬nite dimensional.

1.2. Projective spaces 3

For the most part, we will use left vector spaces. In this case, it is natural

to represent a vector by the row tuple of its coordinates with respect to some

basis; scalar multiplication is a special case of matrix multiplication. If the vector

space has dimension n, then vector space endomorphisms are represented by n n %

matrices, acting on the right, in the usual way:

‘ viAi j

¨ ¥

vA

i

¨ && ¥

if v v vn .

1

The dual space V of a (left) vector space V is the set of linear maps from V

'

to F, with pointwise addition and with scalar multiplication de¬ned by

¨¥

cv

f f¥ cv¨ (

Note that this de¬nition makes V a right vector space.

'

1.2 Projective spaces

A projective space of dimension n over a ¬eld F (not necessarily commuta-

tive!) can be constructed in either of two ways: by adding a hyperplane at in¬nity

to an af¬ne space, or by “projection” of an n 1¨ -dimensional space. Both meth-

¥

ods have their importance, but the second is the more natural.

Thus, let V be an n 1¨ -dimensional left vector space over F. The projective

¥

space PG¥ n F is the geometry whose points, lines, planes, . . . are the vector

¨

subspaces of V of dimensions 1, 2, 3, . . . .

Note that the word “geometry” is not de¬ned here; the properties which are

regarded as geometrical will emerge during the discussion.

Note also the dimension shift: a d-dimensional projective subspace (or ¬‚at)

is a d 1¨ -dimensional vector subspace. This is done in order to ensure that

¥

familiar geometrical properties hold. For example, two points lie on a unique

line; two intersecting lines lie in a unique plane; and so on. Moreover, any d-

dimensional projective subspace is a d-dimensional projective space in its own

right (when equipped with the subspaces it contains).

To avoid confusion (if possible), I will from now on reserve the term rank (in

symbols, rk) for vector space dimension, so that unquali¬ed “dimension” will be

geometric dimension.

A hyperplane is a subspace of codimension 1 (that is, of dimension one less

than the whole space). If H is a hyperplane and L a line not contained in H, then

H L is a point.

)

4 1. Projective spaces

A projective plane (that is, PG¥ 2 F ) has the property that any two lines meet

¨

¨ ¨ ¨

in a (unique) point. For, if rk¥ V 3 and U W V with rk¥ U rk¥ W 2,

0

¨ )

then U W V , and so rk¥ U W 1; that is, U W is a point. From this, we

)

deduce:

Proposition 1.3 (Veblen™s Axiom) If a line intersects two sides of a triangle but

doesn™t contain their intersection, then it intersects the third side also.

33 11 1

31

33

2211

22 323 3 2 2

11

2 3 23

11

2223

1 22

Figure 1.1: Veblen™s Axiom

For the triangle is contained in a plane, and the hypotheses guarantee that the

line in question is spanned by points in the plane, and hence also lies in the plane.

Veblen™s axiom is sometimes called the Veblen-Young Axiom or Pasch™s Ax-

iom. The latter name is not strictly accurate: Pasch was concerned with real pro-

jective space, and the fact that if two intersections are inside the triangle, the third

is outside; this is a property involving order, going beyond the incidence geometry

which is our concern here. In Section 3.1 we will see why 1.3 is referred to as an

“axiom”.

Another general geometric property of projective spaces is the following.

Proposition 1.4 (Desargues™ Theorem) In Figure 1.2, the three points p q r are

collinear.

In the case where the ¬gure is not contained in a plane, the result is obvious

geometrically. For each of the three points p q r lies in both the planes a1 b1 c1 and

a2 b2 c2 ; these planes are distinct, and both lie in the 3-dimensional space spanned

by the three lines through o, and so their intersection is a line.

1.2. Projective spaces 5

o

9 9 88 8 1 1

981

99 9 @ 8 8 4 1 4 A 1 4

c1

b1

99 @ @ A 8A 8 A 1 1 4 4 4 4 4 4

99@ A @ A 8 8 1 1

444 a1

4444 A @ A@ 9 11

4444 88

A

A6 6 6 7 79

5 5 45 11

6 6 7 6 7 6 6 6 88

p q r

55555 a2

7 7 6 8 6 6 6 6 6 16 5

5555 7 87 55 5 5 5 5

b2

c2

Figure 1.2: Desargues™ Theorem

The case where the ¬gure is contained in a plane can be deduced from the

“general” case as follows. Given a point o and a hyperplane H,write aaB bbB if C

oaaB obbB are collinear triples and the lines ab and aB bB intersect in H (but none of

the points a aB b bB lies in H). Now Desargues™ Theorem is the assertion that the

relation is transitive. (For p q r are collinear if and only if every hyperplane

C

containing p and q also contains r; it is enough to assume this for the hyperplanes

not containing the points a aB , etc.) So suppose that aaB bbB ccB . The geomet-

C C

ric argument of the preceding paragraph shows that aaB ccB if the con¬guration C

is not coplanar; so suppose it is. Let od be a line not in this plane, with d H, and D¢

choose dB such that aaB ddB . Then bbB ddB , ccB ddB , and aaB ccB follow in

E

C F

C

C F

C

turn from the non-planar Desargues™ Theorem.

(If we are only given a plane initially, the crucial fact is that the plane can be

embedded in a 3-dimensional space.)

G G

Remark The case where F 2 is not covered by this argument ” can you see

why? ” and, indeed, the projective plane over GF¥ 2¨ contains no non-degenerate

Desargues con¬guration: it only contains seven points! Nevertheless, Desargues™

Theorem holds, in the sense that any meaningful degeneration of it is true in the

projective plane over GF¥ 2¨ . We will not make an exception of this case.

It is also possible to prove Desargues™ Theorem algebraically, by choosing

6 1. Projective spaces

coordinates (see Exercise 1). However, it is important for later developments to

know that a purely geometric proof is possible.

Let V be a vector space of rank n 1 over F, and V its dual space. As we '

saw, V is a right vector space over F, and so can be regarded as a left vector

'

space over the opposite ¬eld F . It has the same rank as V if this is ¬nite. Thus

$

we have projective spaces PG n F and PG¥ n F , standing in a dual relation to

¨¥ H$

¨

one another. More precisely, we have a bijection between the ¬‚ats of PG n F ¨¥

and those of PG¥ n F , given by

¨$

( P¨ ¢ ¦§¥ ' ¢ ¡ ¨

U Ann¥ U f V: u U fu 0¨

I ¥

This correspondence preserves incidence and reverses inclusion:

U1 U2 Ann¥ U2 Ann¥ U1

0 R¨

0 ¨

Q

¨

Ann¥ U1 U2 Ann¥ U1 Ann¥ U2S¨

)

¨

¨

Ann¥ U1 U2 Ann¥ U1 Ann¥ U2

) ¨

H¨

Moreover, the (geometric) dimension of Ann¥ U is n 1 dim¥ U . ¨ ¨

This gives rise to a duality principle, where any con¬guration theorem in pro-

jective space translates into another (over the opposite ¬eld) in which inclusions

are reversed and dimensions suitably modi¬ed. For example, in the plane, the dual

of the statement that two points lie on a unique line is the statement that two lines

meet in a unique point.

We turn brie¬‚y to af¬ne spaces. The description closest to that of projective

spaces runs as follows. Let V be a vector space of rank n over F. The points, lines,

planes, . . . of the af¬ne space AG¥ n F are the cosets of the vector subspaces of

¨

rank 0, 1, 2, . . . . (No dimension shift this time!) In particular, points are cosets of

the zero subspace, in other words, singletons, and we can identify them with the

vectors of V . So the af¬ne space is “a vector space with no distinguished origin”.

The other description is: AG¥ n F is obtained from PG¥ n F by deleting a

¨ ¨

hyperplane together with all the subspaces it contains.

The two descriptions are matched up as follows. Take the vector space

¨ && ¥ ¡

F nT 1

V x x1 xn : x0 xn F

¢ &&

0

Let W be the hyperplane de¬ned by the equation x0 0. The points remaining

D

are rank 1 subspaces spanned by vectors with x0 0; each point has a unique

spanning vector with x0 1. Then the correspondence between points in the two

descriptions is given by U

1 x1 xn x xn

H¨ && ¥ ¤&¨ && ¥

I V

1

1.2. Projective spaces 7

(See Exercise 2.)

In AG¥ n F , we say that two subspaces are parallel if (in the ¬rst description)

¨

they are cosets of the same vector subspace, or (in the second description) they

have the same intersection with the deleted hyperplane. Parallelism is an equiv-

alence relation. Now the projective space can be recovered from the af¬ne space

as follows. To each parallel class of d-dimensional subspaces of AG¥ n F cor-¨

responds a unique d 1¨ -dimensional subspace of PG¥ n 1 F . Adjoin to the

¥ ¨

af¬ne space the points (and subspaces) of PG¥ n 1 F , and adjoin to all mem-

¨

bers of a parallel class all the points in the corresponding subspace. The result is

PG¥ n F .

¨

The distinguished hyperplane is called the hyperplane at in¬nity or ideal hy-

perplane. Thus, an af¬ne space can also be regarded as “a projective space with a

distinguished hyperplane”.

The study of projective geometry is in a sense the outgrowth of the Renais-

sance theory of perspective. If a painter, with his eye at the origin of Euclidean 3-

space, wishes to represent what he sees on a picture plane, then each line through

the origin (i.e., each rank 1 subspace) should be represented by a point of the pic-

ture plane, viz., the point at which it intersects the picture plane. Of course, lines

parallel to the picture plane do not intersect it, and must be regarded as meeting it

in ideal “points at in¬nity”. Thus, the physical picture plane is an af¬ne plane, and

is extended to a projective plane; and the points of the projective plane are in one-

to-one correspondence with the rank 1 subspaces of Euclidean 3-space. It is easily

checked that lines of the picture plane correspond to rank 2 subspaces, provided

we make the convention that the points at in¬nity comprise a single line. Not that

the picture plane really is af¬ne rather than Euclidean; the ordinary distances in it

do not correspond to distances in the real world.

Exercises

1. Prove Desargues™ Theorem in coordinates.

2. Show that the correspondence de¬ned in the text between the two descrip-

tions of af¬ne space is a bijection which preserves incidence, dimension, and par-

allelism.

3. The L TEX typesetting system provides facilities for drawing diagrams. In

A

a diagram, the slope of a line is restricted to being in¬nity or a rational number

whose numerator and denominator are each at most 6 in absolute value.

(a) What is the relation between the slopes of the six lines of a complete

quadrangle (all lines joining four points)? Investigate how such a ¬gure can be

8 1. Projective spaces

drawn with the above restriction on the slopes.

(b) Investigate similarly how to draw a Desargues con¬guration.

1.3 The “Fundamental Theorem of Projective Geometry”

An isomorphism between two projective spaces is a bijection between the

point sets of the spaces which maps any subspace into a subspace (when applied in

either direction). A collineation of PG n F is an isomorphism from PG¥ n F to

¨¥ ¨

itself. The theorem of the title of this section has two consequences: ¬rst, that iso-

morphic projective spaces have the same dimension and the same coordinatising

¬eld; second, a determination of the group of all collineations.

We must assume that n 1; for the only proper subspaces of a projective line

W

are its points, and so any bijection is an isomorphism, and the collineation group

is the full symmetric group. (There are methods for assigning additional structure

to a projective line, for example, using cross-ratio; these will be discussed later

on, in Section 4.5.)

The general linear group GL¥ n 1 F is the group of all non-singular linear

¨

transformations of V F nT 1 ; it is isomorphic to the group of invertible n 1¨

X ¥

%

n 1¨ matrices over F. (In general, the determinant is not well-de¬ned, so we

¥

cannot identify the invertible matrices with those having non-zero determinant.)

Any element of GL¥ n 1 F maps subspaces of V into subspaces of the same

¨

rank, and preserves inclusion; so it induces a collineation of PG¥ n F . The group

¨

Aut¥ F of automorphisms of F has a coordinate-wise action on V nT 1 ; these trans-

¨

formations also induce collineations. The group generated by GL¥ n 1 F and ¨

Aut¥ F (which is actually their semi-direct product) is denoted by “L¥ n 1 F ;

¨ ¨

its elements are called semilinear transformations. The groups of collineations of

PG¥ n F induced by GL¥ n 1 F and “L¥ n 1 F are denoted by PGL¥ n 1 F

¨ ¨ ¨ ¨

and P“L¥ n 1 F , respectively.

¨

More generally, a semi-linear transformation from one vector space to another

is the composition of a linear transformation and a coordinate-wise ¬eld automor-

phism of the target space.

Theorem 1.5 (Fundamental Theorem of Projective Geometry) Any isomorphism

between projective spaces of dimension at least 2 is induced by a semilinear trans-

formation between the underlying vector spaces, unique up to scalar multiplica-

tion.

1.3. The “Fundamental Theorem of Projective Geometry” 9

Before outlining the proof, we will see the two important corollaries of this

result. Both follow immediately from the theorem (in the second case, by taking

the two projective spaces to be the same).

Corollary 1.6 Isomorphic projective spaces of dimension at least 2 have the same

dimension and are coordinatised by isomorphic ¬elds.

Corollary 1.7 (a) For n 1, the collineation group of PG¥ n F is the group

W ¨

P“L¥ n 1 F .

¨

(b) The kernel of the action of “L¥ n 1 F on PG¥ n F is the group of non-zero

¨ ¨

scalars (acting by left multiplication).

Remark The point of the theorem, and the reason for its name, is that the alge-

braic structure of the underlying vector space can be recovered from the incidence

geometry of the projective space. The proof is a good warm-up for the coor-

dinatisation theorems I will be discussing soon. In fact, the proof concentrates

on Corollary 1.7, for ease of exposition. The dimension of a projective space is

two less than the number of subspaces in a maximal chain (under inclusion); and

our argument shows that the geometry determines the coordinatising ¬eld up to

isomorphism.

Proof We show ¬rst that two semi-linear transformations which induce the same

collineation differ only by a scalar factor. By following one by the inverse of

the other, we see that it suf¬ces to show that a semi-linear transformation which

¬xes every point of PG¥ n F is a scalar multiplication. So let v vσ A ¬x every

¨

point of PG n F , where σ Aut¥ F and A GL¥ n 1 F . Then every vector

¨¥ ¢ ¨ ¢

¨

is mapped to a scalar multiple of itself. Let e0 en be the standard basis for V .

Y&

Then (since σ ¬xes the standard basis vectors) we have ei A »i ei for i 0 &&

n.

Also,

»0 e0 »n en

e en A

&& ¥

¨ &&

0

»¥ e0

en say,

Y&

¨

so »0 »n ».

&Y

» µσ » 0

Now, for any µ F, the vector 1 µ0 0¨ is mapped to the vector 0¨ ;

¢ && ¥

¥ &Y

so we have »µ µσ ». Thus

vσ A vσ » »v

10 1. Projective spaces

for any vector v, as required.

Note that the ¬eld automorphism σ is conjugation by the element » (that is,

µσ »µ»` 1 ); in other words, an inner automorphism.

Now we prove that any isomorphism is semilinear. The strategy is similar.

Call an n 2¨ tuple of points special if no n 1 of them are linearly dependent.

¥

We have:

There is a linear map carrying any special tuple to any other (in the

same space, or another space of the same dimension over the same

¬eld).

(For, given a special tuple in the ¬rst space, spanning vectors for the ¬rst n 1

points form a basis e0 en , and the last point is spanned by a vector with all

&&

coordinates non-zero relative to this basis. Adjusting the basis vectors by scalar

factors, we may assume that the last point is spanned by e0 en . Similarly,

&&

the points of a special tuple in the second space are spanned by the vectors of a

basis f0 fn , and f0 fn . The unique linear transformation carrying the

&&

&&

¬rst basis to the second also carries the ¬rst special tuple to the second.)

Let θ be any isomorphism. Then there is a linear map φ which mimics the

effect of θ on a special n 2¨ -tuple. Composing θ with the inverse of φ, we

¥

obtain an automorphism of PG¥ n F which ¬xes the n 2¨ -tuple pointwise. We

¨ ¥

have to show that such an automorphism is the product of a scalar and a ¬eld

automorphism. (Note that, as we saw above, left and right multiplications by »

differ by an inner automorphism.)

We assume that n 2; this simpli¬es the argument, while retaining its essential

features. So let g be a collineation ¬xing the spans of e0 e1 e2 and e0 e1 e2 . We

use homogeneous coordinates, writing these vectors as 1 0 0¨ , 0 1 0¨ , 0 0 1¨ ,

¥ ¥ ¥

and 1 1 1¨ , and denote the general point by x y z¨ .

¥ ¥

a¨ ¥ ¡

The points on the line 0 0 x2 , apart from

x 1 0 0¨ , have the form x 0 1¨

¥ ¥

for x F, and so can be identi¬ed with elements of F. Now the bijection be-

¢

¥¡

tween this set and the set of points 0 y 1¨ on the line 0 x1 x2 , given by

¥ (¨

x 0 1¨ 0 x 1¨ , can be geometrically de¬ned in a way which is invariant un-

¥ b ¥

der collineations ¬xing the four reference points (see Fig. 1.3). The ¬gure also

shows that the coordinates of all points in the plane are determined.

Furthermore, the operations of addition and multiplication in F can be de¬ned

geometrically in the same sense (see Figures 1.4 and 1.6). (The de¬nitions look

( ¥¡

more familiar if we take the line 1 x2 0¨ to be at in¬nity, and draw the ¬gure

x

in the af¬ne plane with lines through 1 0 0¨ and 0 1 0¨ horizontal and vertical

¥ ¥

respectively. this has been done for addition in Figure 1.5; the reader should draw

1.3. The “Fundamental Theorem of Projective Geometry” 11

93c g

0h 1h 0i

c 3 c3 99

c39

c

3 99 c

3 c3

99 c

c 33

99

ff g c c 33

0h xh 1i

f 99f g cd

c d 33 f f f 1h 1h 0i

99 9 c

dg d 3 f f

cc

f 3f 3 d d 9 xh xh 1i

f f33 cc ff

d d 99

ccfff

e ee g 3 3 e e e e e g 9 9e 9 d e

0h 1h 1i

1h 1h 1i

cf fc f f e e e e e 3

dd 3e e e

99 c

d 3

e

g fe e e e

g g g

0h 0h 1i 1h 0h 1i xh 0h 1i 1h 0h 0i

Figure 1.3: Bijection between the axes

the corresponding diagram for multiplication.) It follows that any collineation

¬xing our four basic points induces an automorphism of the ¬eld F, and its actions

on the coordinates agree. The theorem is proved.

A group G acting on a set „¦ is said to be t-transitive if, given any two t-tuples

±t and β 1 βt of distinct elements of „¦, some element of G carries

¨ &Y ¥

¨ &Y ¥

±1

the ¬rst tuple to the second. G is sharply t-transitive if there is a unique such

element. (If the action is not faithful, it is better to say: two elements of G which

agree on t distinct points of „¦ agree everywhere.)

Since any two distinct points of PG¥ n F are linearly independent, we see that

¨

P“L¥ n 1 F (or even PGL¥ n 1 F ) is 2-transitive on the points of PG¥ n F . It

¨ ¨ ¨

is never 3-transitive (for n 1); for some triples of points are collinear and others

W

are not, and no collineation can map one type to the other.

12 1. Projective spaces

pqc g

0h 1h 0i

c q c qp p

c

c q q pp

c

c q q pp

c qp

c q pp

cc q q pp

g rcd qp

1h 1h 0i

crd

c r d qq p

cc r r d q qd pp p

cc rr q q d d p

g 5p d5 5 5 5 5

c

g r5 r q q 5 5 5 5 p d d

yh yh 1i

c xT yh yh 1i

c c 5 5 5 5 5 5 5 5 q q rr pp p dd

c5 qr

g5 5 5 g

g

gg

0h 0h 1i yh 0h 1i xh 0h 1i xT yh 0h 1i 1h 0h 0i

Figure 1.4: Addition

I will digress here to describe the analogous situation for PG¥ 1 F , even ¨

though the FTPG does not apply in this case.

Proposition 1.8 (a) The group PGL¥ 2 F is 3-transitive on the points of PG¥ 1 F ,

¨ ¨

and is sharply 3-transitive if and only if F is commutative.

(b) There exist skew ¬elds F for which the group PGL 2 F is 4-transitive on ¨¥

PG¥ 1 F .

¨

Proof The ¬rst part follows just as in the proof of the FTPG, since any three

points of PG¥ 1 F have the property that no two are linearly dependent. Again,

¨

as in that theorem, the stabiliser of the three points with coordinates 1 0¨ , 0 1¨ ¥ ¥

and 1 1¨ is the group of inner automorphisms of F, and so is trivial if and only if

¥

F is commutative.

1.3. The “Fundamental Theorem of Projective Geometry” 13

gd gd

yh yi xT yh yi

d d

dd dd

g g g g

0h 0i yh 0i xh 0i xT yh 0i

Figure 1.5: Af¬ne addition

There exist skew ¬elds F with the property that any two elements different

from 0 and 1 are conjugate in the multiplicative group of F. Clearly these have

the required property. (This fact is due to P. M. Cohn [15]; it is established by

a construction analogous to that of Higman, Neumann and Neumann [20] for

groups. Higman et al. used their construction to show that there exist groups

in which all non-identity elements are conjugate; Cohn™s work shows that there

are multiplicative groups of skew ¬elds with this property. Note that such a ¬eld

D

has characteristic 2. For, if not, then 1 1 0, and any automorphism must ¬x

1 1.)

Finally, we consider collineations of af¬ne spaces.

Parallelism in an af¬ne space has an intrinsic, geometric de¬nition. For two d-

¬‚ats are parallel if and only if they are disjoint and some d 1¨ -¬‚at contains both. ¥

It follows that any collineation of AG¥ n F preserves parallelism. The hyperplane

¨

at in¬nity can be constructed from the parallel classes (as we saw in Section 1.2);

so any collineation of AG¥ n F induces a collineation of this hyperplane, and

¨

hence of the embedding PG¥ n F . Hence:

¨

Theorem 1.9 The collineation group of AG¥ n F is the stabiliser of a hyperplane

¨

in the collineation group of PG¥ n F . ¨

Using this, it is possible to determine the structure of this group for n 1 (see

W

Exercise 2).

Proposition 1.10 For n 1, the collineation group of AG¥ n F is the semi-direct

W ¨

product of the additive group of F n and “L¥ n F . ¨

This group is denoted by A“L¥ n F . The additive group acts by translation,

¨

and the semilinear group in the natural way.

14 1. Projective spaces

9v@wc g

0h 1h 0i

c w @ w c @v 9v 9

c w c@ w @ v v 9 9 u

c w c w@ @ v u v 9 9 u

g sc w @ u uv 9 1h yh 0i

csw @u v 9

cc

w g u @ s @s u v v 9 9 xh xyh 1i

wu @ s v 9 g cd w 1h 1h 0i

c g d uw u

@ s sv v 99 c

uw @d@

xyh xyh 1i

c uc u

d d v ss 9 u

w w@@

dg v vd t gt 9 9s t s t t t c uc u w w

1h yh 1i

t t v t v gd 9 9 d s s @

yh yh 1i

cu uc u w w t @

@ttt

vv 9 9 d s t

1h 1h 1i

cu uc t t wt w t t @ t @

d s ds

vv 9 9 ctw @

g ut t

g

g

g gg

0h 0h 1i 1h 0h 1i yh 0h 1i xh 0h 1i xyh 0h 1i 1h 0h 0i

Figure 1.6: Multiplication

Exercises

1. Prove the FTPG for n 2. W

2. Use the correspondence between the two de¬nitions of AG¥ n F given in ¨

the last section to deduce Proposition 1.10 from Theorem 1.9.

1.4 Finite projective spaces

Over the ¬nite ¬eld GF¥ q¨ , the n-dimensional projective and af¬ne spaces

and their collineation groups are ¬nite, and can be counted. In this section, we

display some of the relevant formul¦. We abbreviate PG¥ n GF¥ q¨ to PG¥ n q¨ , &

¨

and similarly for af¬ne spaces, collineation groups, etc.

A vector space of rank n over GF¥ q¨ is isomorphic to GF¥ q¨ n , and so the

number of vectors is qn . In consequence, the number of vectors outside a subspace

1.4. Finite projective spaces 15

of rank k is qn qk .

Proposition 1.11 The number of subspaces of rank k in a vector space of rank n

over GF¥ q¨ is

q n 1¨ qn q¨ qn qk 1

yY&S ¥x ¥

¥©©© ¨¨ ``

q k 1¨ qk q¨ qk qk 1

y&&S ¥x ¥

¥©©©

n

Remark This number is called a Gaussian coef¬cient, and is denoted by k q .

Proof First we count the number of choices of k linearly independent vectors.

The ith vector may be chosen arbitrarily outside the subspace of rank i 1 spanned

by its predecessors, hence in qn qi 1 ways. Thus, the numerator is the required

`

number of choices.

Now any k linearly independent vectors span a unique subspace of rank k; so

the number of subspaces is found by dividing the number just calculated by the

number of choices of a basis for a space of rank k. But the latter is given by the

same formula, with k replacing n.

Proposition 1.12 The order of GL¥ n q¨ is

n

1¨ qn qn qn 1

q q¨ H¨ `

y&&S ¥x ¥

¥©©©

The order of “L¥ n q¨ is the above number multiplied by d, where q pd with p

prime; and the orders of PGL¥ n q¨ and P“L¥ n q¨ are obtained by dividing these

numbers by q 1¨ .

¥

Proof An element of GL¥ n q¨ is uniquely determined by the image of the stan-

dard basis, which is an arbitrary basis of GF¥ q¨ n; and the proof of Proposition 1.11

shows that the number of bases is the number quoted. The remainder of the propo-

sition follows from the remarks in Section 1.3, since GF¥ q¨ has q 1 non-zero

scalars, and its automorphism group has order d.

The formula for the Gaussian coef¬cient makes sense, not just for prime power

values of q, but for any value of q different from 1. There is a combinatorial

interpretation for any integer q 1 (Exercise 3). Moreover, by l™Hˆ pital™s rule,

o

W

limq‚ 1 q a 1¨ qb 1¨ a b; it follows that

„& ¥

¥

… n n

lim

k k

q‚ 1 q

16 1. Projective spaces

This illustrates just one of the many ways in which subspaces of ¬nite vector

spaces resemble subsets of sets.

It follows immediately from Propopsition 1.11 that the numbers of k-dimensional

¬‚ats in PG¥ n q¨ and AG¥ n q¨ are nT 1 q and qn k n q respectively.

1

`

kT k

Projective and af¬ne spaces provide important examples of designs, whose

parameters can be expressed in terms of the Gaussian coef¬cients.

A t-design with parameters v k »¨ , or t-¥ v k »¨ design, consists of a set X

¥

of v points, and a collection of k-element subsets of X called blocks, with the

property that any t distinct points of X are contained in exactly » blocks. Designs

were ¬rst used by statisticians, such as R. A. Fisher, for experimental design (e.g.

to facilitate analysis of variance). The terms “design” and “block”, and the letter

v (the initial letter of “variety”), re¬‚ect this origin.

Proposition 1.13 (a) The points and m-dimensional ¬‚ats in PG¥ n q¨ form a 2-

design with parameters

…‘ … …

n 1 m 1 n1

1 1 m 1

q q q’

(b) The points and m-dimensional ¬‚ats of AG¥ n q¨ form a 2-design with param-

eters ‘

…

n1

qn qm

m 1 q’

2, then it is a 3-design, with »

`` n2

If q m 2 2 .

Proof The values of v and k are clear in both cases.

(a) Let V be the underlying vector space of rank n 1. We want to count

the subspaces of rank m 1 containing two given rank 1 subspaces P1 and P2 . If

L P1 P2 , then L has rank 2, and a subspace contains P1 and P2 if and only if it

contains L. Now, by the Third Isomorphism Theorem, the rank m 1 subspaces

containing L are in 1-1 correspondence with the rank m 1 subspaces of the rank

n 1 space V L.

(b) In AG¥ n q¨ , to count subspaces containing two points, we may assume

(by translation) that one of the points is the origin. An af¬ne ¬‚at containing the

origin is a vector subspace, and a subspace contains a non-zero vector if and only

if it contains the rank 1 subspace it spans. The result follows as before. In the

case when q 2, a rank 1 subspace contains only one non-zero vector, so any two

distinct non-zero vectors span a rank 2 subspace.

1.4. Finite projective spaces 17

Remark The essence of the proof is that the quotient of either PG¥ n q¨ or

AG¥ n q¨ by a ¬‚at F of dimension d is PG n d 1 q¨ . (The ¬‚ats of the quotient

¥

space are precisely the ¬‚ats of the original space containing F.) This assertion is

true over any ¬eld at all, and lies at the basis of an approach to geometry which

we will consider in Chapter 5.

An automorphism of a design is a permutation of the points which maps any

block to a block.

Proposition 1.14 For 0 m n, the design of points and m-dimensional ¬‚ats in

“ “

PG¥ n q¨ or AG¥ n q¨ is P“L n 1 q¨ or A“L¥ n 1 q¨ respectively, except in the

¥

af¬ne case with q 2 and m 1.

Proof By the results of Section 1.3, it suf¬ces to show that the entire geometry

can be recovered from the points and m-dimensional ¬‚ats. This follows immedi-

ately from two observations:

(a) the unique line containing two points is the intersection of all the m-dimensional

¬‚ats containing them;

(b) except for af¬ne spaces over GF¥ 2¨ , a set of points is a ¬‚at if and only if it

contains the line through any two of its points.

Af¬ne spaces over GF¥ 2¨ are exceptional: lines have just two points, and any two

points form a line. However, analogous statements hold for planes: three points

lie in a unique plane, and we have

(aa) the plane through three points is the intersection of all the ¬‚ats of dimen-

sion m which contain them (for m 1); W

(bb) a set of points is a ¬‚at if and only if it contains the plane through any three

of its points.

The proofs are left as exercises.

Exercises

1. Prove the assertions (a), (b), (aa), (bb) in Proposition 1.14.

2. Prove that the probability that a random n n matrix over a given ¬nite ¬eld

%

GF¥ q¨ is non-singular tends to a limit c¥ q¨ as n ∞, where 0 c¥ q¨ 1.

“ “R

18 1. Projective spaces

3. Prove that the total number F n of subspaces of a vector space of rank n

¨¥

over a given ¬nite ¬eld GF¥ q¨ satis¬es the recurrence

n

Fn 1¨ 2F n q 1¨ F n 1¨

¥ ¥ ¨ ¥ H ¥

4. Let S be an “alphabet” of size q, with two distinguished elements 0 and 1

(but not necessarily a ¬nite ¬eld). A k n matrix with entries from S is (as usual)

%

in reduced echelon form if

” it has no zero rows;

” the ¬rst non-zero entry in any row is a 1;

” the “leading 1s” in later rows occur further to the right;

” the other entries in the column of a “leading 1” are all 0.

n

Prove that the number of k n matrices in reduced echelon form is k q . Verify in

%

detail in the case n 4, k 2.

5. Use the result of Exercise 4 to prove the recurrence relation

… … …

n n 1 n 1

qn

k k k 1

q q q

2

Projective planes

Projective and af¬ne planes are more than just spaces of smallest (non-trivial)

dimension: as we will see, they are truly exceptional, and also they play a crucial

rˆ le in the coordinatisation of arbitrary spaces.

o

2.1 Projective planes

We have seen in Sections 1.2 and 1.3 that, for any ¬eld F, the geometry

PG 2¡ F has the following properties:

¢

(PP1) Any two points lie on exactly one line.

(PP2) Any two lines meet in exactly one point.

(PP3) There exist four points, no three of which are collinear.

I will now use the term projective plane in a more general sense, to refer to any

structure of points and lines which satis¬es conditions (PP1)-(PP3) above.

In a projective plane, let p and L be a point and line which are not incident.

The incidence de¬nes a bijection between the points on L and the lines through p.

By (PP3), given any two lines, there is a point incident with neither; so the two

lines contain equally many points. Similarly, each point lies on the same number

of lines; and these two constants are equal. The order of the plane is de¬ned to

be one less than this number. The order of PG 2¡ F is equal to the cardinality of

¢

F. (We saw in the last section that a projective line over GF q¢ has 2 q q 1

£ ¦

¥

1¤

points; so PG 2¡ q¢ is a projective plane of order q. In the in¬nite case, the claim

follows by simple cardinal arithmetic.)

19

20 2. Projective planes

Given a ¬nite projective plane of order n, each of the n 1 lines through a point

¦

p contains n further points, with no duplications, and all points are accounted for

in this way. So there are n2 n 1 points, and the same number of lines. The

¦ ¦

points and lines form a 2- n2 n 1¡ n 1¡ 1¢ design. The converse is also true

¦¦ ¦

(see Exercise 2).

Do there exist projective planes not of the form PG 2¡ F ? The easiest such

¢

examples are in¬nite; I give two completely different ones below. Finite examples

will appear later.

Example 1: Free planes. Start with any con¬guration of points and lines having

the property that two points lie on at most one line (and dually), and satisfying

(PP3). Perform the following construction. At odd-numbered stages, introduce

a new line incident with each pair of points not already incident with a line. At

even-numbered stages, act dually: add a new point incident with each pair of

lines for which such a point doesn™t yet exist. After countably many stages, a

projective plane is obtained. For given any two points, there will be an earlier

stage at which both are introduced; by the next stage, a unique line is incident with

both; and no further line incident with both is added subsequently; so (PP1) holds.

Dually, (PP2) holds. Finally, (PP3) is true initially and remains so. If we start

with a con¬guration violating Desargues™ Theorem (for example, the Desargues

con¬guration with the line pqr “broken” into separate lines pq, qr, rp), then the

resulting plane doesn™t satisfy Desargues™ Theorem, and so is not a PG 2¡ F . ¢

Example 2: Moulton planes. Take the ordinary real af¬ne plane. Imagine that

the lower half-plane is a refracting medium which bends lines of positive slope

so that the part below the axis has twice the slope of the part above, while lines

with negative (or zero or in¬nite) slope are unaffected. This is an af¬ne plane, and

has a unique completion to a projective plane (see later). The resulting projective

plane fails Desargues™ theorem. To see this, draw a Desargues con¬guration in the

ordinary plane in such a way that just one of its ten points lies below the axis, and

just one line through this point has positive slope.

The ¬rst examples of ¬nite planes in which Desargues™ Theorem fails were

constructed by Veblen and Wedderburn [38]. Many others have been found since,

but all known examples have prime power order. The Bruck“Ryser Theorem [4]

asserts that, if a projective plane of order n exists, where n 1 or 2 (mod 4), then

§

n must be the sum of two squares. Thus, for example, there is no projective plane

of order 6 or 14. This theorem gives no information about 10, 12, 15, 18, . . . .

2.1. Projective planes 21

Recently, Lam, Swiercz and Thiel [21] showed by an extensive computation that

there is no projective plane of order 10. The other values mentioned are undecided.

An af¬ne plane is an incidence structure of points and lines satisfying the

following conditions (in which two lines are called parallel if they are equal or

disjoint):

(AP1) Two points lie on a unique line.

(AP2) Given a point p and line L, there is a unique line which contains p and is

parallel to L.

(AP3) There exist three non-collinear points.

Remark. Axiom (AP2) for the real plane is an equivalent form of Euclid™s “par-

allel postulate”. It is called “Playfair™s Axiom”, although it was stated explicitly

by Proclus.

Again it holds that AG 2¡ F is an af¬ne plane. More generally, if a line and all

¢

its points are removed from a projective plane, the result is an af¬ne plane. (The

removed points and line are said to be “at in¬nity”. Two lines are parallel if and

only if they contain the same point at in¬nity.

Conversely, let an af¬ne plane be given, with point set and line set . It

¨ ©

follows from (AP2) that parallelism is an equivalence relation on . Let be the ©

set of equivalence classes. For each line L , let L L Q , where Q is the

©

¥

parallel class containing L. Then the structure with point set , and line set

¨

L : L , is a projective plane. Choosing as the line at in¬nity, we

"!

©

recover the original af¬ne plane.

We will have more to say about af¬ne planes in Section 3.5.

Exercises

1. Show that a structure which satis¬es (PP1) and (PP2) but not (PP3) must be

of one of the following types:

(a) There is a line incident with all points. Any further line is a singleton,

repeated an arbitrary number of times.

(b) There is a line incident with all points except one. The remaining lines all

contain two points, the omitted point and one of the others.

¦¦

2. Show that a 2- n2 n 1¡ n 1¡ 1¢ design (with n 1) is a projective plane

¦ #

of order n.

3. Show that, in a ¬nite af¬ne plane, there is an integer n 1 such that#

22 2. Projective planes

$ every line has n points;

$ every point lies on n 1 lines;

¦

$ there are n2 points;

$ there are n 1 parallel classes with n lines in each.

¦

(The number n is the order of the af¬ne plane.)

4. (The Friendship Theorem.) In a ¬nite society, any two individuals have a

unique common friend. Prove that there exists someone who is everyone else™s

friend. ©

[Let X be the set of individuals, F x¢ : x X , where F x¢ is the set

¥

©%¡

of friends of X . Prove that, in any counterexample to the theorem, X is a ¢

projective plane, of order n, say.

Now let A be the real matrix of order n2 n 1, with x¡ y¢ entry 1 if x and y¦¦

are friends, 0 otherwise. Prove that

A2 nI J¡

¦

¥

where I is the identity matrix and J the all-1 matrix. Hence show that the real

symmetric matrix A has eigenvalues n 1 (with multiplicity 1) and n. Using

¦ (&

'

the fact that A has trace 0, calculate the multiplicity of the eigenvalue n, and '

hence show that n 1.] ¥

5. Show that any Desargues con¬guration in a free projective plane must lie

within the starting con¬guration. [Hint: Suppose not, and consider the last point

or line to be added.]

2.2 Desarguesian and Pappian planes

It is no coincidence that we distinguished the free and Moulton planes from

PG 2¡ F s in the last section by the failure of Desargues™ Theorem.

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Theorem 2.1 A projective plane is isomorphic to PG 2¡ F for some F if and only ¢

if it satis¬es Desargues™ Theorem.

I do not propose to give a detailed proof of this important result; but some

comments on the proof are in order.

We saw in Section 1.3 that, in PG 2¡ F , the ¬eld operations (addition and mul-

¢

tiplication) can be de¬ned geometrically, once a set of four points with no three

2.2. Desarguesian and Pappian planes 23

collinear has been chosen. By (PP3), such a set of points exists in any projective

plane. So it is possible to de¬ne two binary operations on a set consisting of a

line with a point removed, and to coordinatise the plane with this algebraic ob-

ject. Now it is obvious that any ¬eld axiom translates into a certain “con¬guration

theorem”, so that the plane is a PG 2¡ F if and only if all these “con¬guration

¢

theorems” hold. What is not obvious, and quite remarkable, is that all these “con-

¬guration theorems” follow from Desargues™ Theorem.

Another method, more dif¬cult in principle but much easier in detail, exploits

the relation between Desargues™ Theorem and collineations.

Let p be a point and L a line. A central collineation with centre p and axis

L is a collineation ¬xing every point on L and every line through p. It is called

an elation if p is on L, a homology otherwise. The central collineations with

centre p and axis L form a group. The plane is said to be p¡ L¢ -transitive if this

group permutes transitively the set M p¡ L M for any line M L on p (or,

0)

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equivalently, the set of lines on q different from L and pq, where q p is a point 2¥

of L).

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