стр. 1(всего 7)СОДЕРЖАНИЕ >>
Projective and Polar Spaces

Peter J. Cameron
Queen Mary and Westп¬Ѓeld 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 вЂњgeometric in-
tuitionвЂќ among our students. On the other hand, recent decades have seen renewed
links between geometry and physics, to the beneп¬Ѓt of both disciplines. One of the
worldвЂ™s leading mathematicians has argued that the insights of вЂњpre-calculusвЂќ ge-
ometry have a rЛ† le to play at all levels of mathematical activity (ArnolвЂ™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 п¬Ѓeld, and unencumbered by metrics or continuity (or even betweenness).
The major themes are the projective and afп¬Ѓne 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 п¬Ѓeld with two elements), and an outline
of the general argument.
The classical works on the subject are the books of DieudonnВґ [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 classiп¬Ѓcation of the п¬Ѓnite simple groups with its many geometric
spin-offs, BuekenhoutвЂ™s geometries associated with diagrams, etc.), and I have
included some material not found in the classical books.
Roughly speaking, the п¬Ѓrst п¬Ѓve chapters are on projective spaces, the last
п¬Ѓve on polar spaces. In more detail: Chapter 1 introduces projective and afп¬Ѓne

i
ii

spaces synthetically, and derives some of their properties. Chapter 2, on projective
planes, discusses the rЛ† le of DesarguesвЂ™ and PappusвЂ™ 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 п¬Ѓnite п¬Ѓelds of odd char-
acteristic (SegreвЂ™s theorem); the structure of projective lines; and generation and
simplicity of the projective special linear groups. Chapter 5 outlines BuekenhoutвЂ™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 reп¬‚exive sesquilinear forms,
and gives the classiп¬Ѓcation of these forms. Chapter 7 deп¬Ѓnes polar spaces, the
geometries associated with such forms, and gives a number of these properties;
the VeldkampвЂ“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 п¬Ѓeld. Chapter 8 discusses two important low-dimensional phenomena,
the Klein quadric and triality, proceeding as far as to deп¬Ѓne the polarity deп¬Ѓning
the SuzukiвЂ“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 afп¬Ѓne spaces, and also having a high degree of symmetry. In the п¬Ѓnal chapter,
we deп¬Ѓne 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-
п¬Ѓeld 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 п¬Ѓeld, 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 п¬Ѓeld; 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 п¬Ѓrst edition which was
published in the QMW Maths Notes series in 1991. I have converted the п¬Ѓles
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 TaylorвЂ™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/Лњfdc/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 вЂњFundamental Theorem of Projective GeometryвЂќ . . . . . . . 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
В

3.1 The GF 2ВЎ case . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 An application . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.4 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.5 Afп¬Ѓne 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 SegreвЂ™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 Classiп¬Ѓcation 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 BuekenhoutвЂ“Shult . . . . . . . . . . . . . . . . . . . . . 105
7.4 Recognising hyperbolic quadrics . . . . . . . . . . . . . . . . . . 107
В

7.5 Recognising quadrics over GF 2ВЎ . . . . . . . . . . . . . . . . . . 109
7.6 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . 113

8 The Klein quadric and triality 115
8.1 The Pfafп¬Ѓan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 afп¬Ѓne 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 вЂњп¬ЃeldвЂќ
will mean вЂњdivision ringвЂќ or вЂњskew п¬ЃeldвЂќ, while the word вЂњcommutativeвЂќ will be
used where necessary. Often, though, I will say вЂњskew п¬ЃeldвЂќ, as a reminder. (Of
course, this refers to the multiplication only; addition will always be commuta-
tive.)
Given a п¬Ѓeld F, let
В  В© ВЁ Вў В§В¦ВҐ В¤Вў ВЎ В  Вў ВЎ В
О± FnО±  В  В©
I n : 0 n : n 1F 0
ВЈ ВЈ
ВЁВҐ В
Then I is an ideal in , hence I c for some non-negative integer c called the
ВЈ
characteristic of F. The characteristic is either 0 or a prime number. For each
value of the characteristic, there is a unique prime п¬Ѓeld which is a subп¬Ѓeld of any
п¬Ѓeld 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 п¬Ѓeld extensions, degree,
and so on.
Much of the time, we will be concerned with п¬Ѓnite п¬Ѓelds. The main results
about these are as follows.

Theorem 1.1 (WedderburnвЂ™s Theorem) A п¬Ѓnite п¬Ѓeld is commutative.

1
2 1. Projective spaces

Theorem 1.2 (GaloisвЂ™ Theorem) A п¬Ѓnite п¬Ѓeld has prime power order. For any
prime power q, there is a unique п¬Ѓnite п¬Ѓeld of order q.
В
The unique п¬Ѓeld of order q is denoted by GFВҐ qВЁ . If q pd with p prime,
its additive structure is that of a d-dimensional vector space over its prime п¬Ѓeld
GFВҐ pВЁ (the integers modulo p). Its multiplicative group is cyclic (of order q 1), 
В
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.
ВЁВҐ
An anti-automorphism of a п¬Ѓeld is a bijection Пѓ with the properties
Пѓ
cПѓ cПѓ
В
c c2
ВҐ ВЁ 
1 
1 2
Пѓ ПѓПѓ
В
c c2 c2 c1
1

The identity (or, indeed, any automorphism) is an anti-automorphism of a com-
mutative п¬Ѓeld. Some non-commutative п¬Ѓelds have anti-automorphisms. A well-
known example is the п¬Ѓeld of quaternions, with a basis over consisting of
 
elements 1 i j k satisfying

В  В  В  В  В  В
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 п¬Ѓeld F is the п¬Ѓeld F , where the binary oper-
!ВЁВ©    ВҐ ВЁ "#   ВҐ
ation is deп¬Ѓned by the rule
"
В
c1 c2 c2 c1 В©
"
Thus, an anti-automorphism of F is just an isomorphism between F and its oppo-
site F . \$
For non-commutative п¬Ѓelds, 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
ВҐВ
vector v as cv, then c1 2 vВЁ
c 1 c2 v holds. In a right vector space, this condition
c ВҐ ВЁ
ВҐВ
reads c1 2 vВЁ
c 2 c1 v. It is more natural to write the scalars on the right (thus:
c
ВҐ ВЁ
В ВЁВҐ
vc), so that the condition is v c2 c1 vВҐ c2 c1 ). A right vector space over F is a ВЁ
left vector space over F . \$
Our vector spaces will almost always be п¬Ѓnite 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:

в€‘ viAi j
В ВЁ ВҐ
vA
i

ВЁ  &&  ВҐ В
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 deп¬Ѓned by
В ВЁВҐ
cv
f fВҐ cvВЁ (

Note that this deп¬Ѓnition makes V a right vector space.
'
1.2 Projective spaces
A projective space of dimension n over a п¬Ѓeld F (not necessarily commuta-
tive!) can be constructed in either of two ways: by adding a hyperplane at inп¬Ѓnity
to an afп¬Ѓne space, or by вЂњprojectionвЂќ of an n 1ВЁ -dimensional space. Both meth-
ВҐ
ods have their importance, but the second is the more natural.
Thus, let V be an n 1ВЁ -dimensional left vector space over F. The projective
ВҐ
space PGВҐ n F is the geometry whose points, lines, planes, . . . are the vector
ВЁ
subspaces of V of dimensions 1, 2, 3, . . . .
Note that the word вЂњgeometryвЂќ is not deп¬Ѓned here; the properties which are
regarded as geometrical will emerge during the discussion.
Note also the dimension shift: a d-dimensional projective subspace (or п¬‚at)
is a d 1ВЁ -dimensional vector subspace. This is done in order to ensure that
ВҐ
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 unqualiп¬Ѓed вЂњdimensionвЂќ 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, PGВҐ 2 F ) has the property that any two lines meet
ВЁ
В ВЁ В ВЁ В ВЁ
in a (unique) point. For, if rkВҐ V 3 and U W V with rkВҐ U rkВҐ W 2,
0
В  В ВЁ )
then U W V , and so rkВҐ U W 1; that is, U W is a point. From this, we
)

deduce:

Proposition 1.3 (VeblenвЂ™s Axiom) If a line intersects two sides of a triangle but
doesnвЂ™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: VeblenвЂ™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.
VeblenвЂ™s axiom is sometimes called the Veblen-Young Axiom or PaschвЂ™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
вЂњaxiomвЂќ.
Another general geometric property of projective spaces is the following.

Proposition 1.4 (DesarguesвЂ™ Theorem) In Figure 1.2, the three points p q r are 
collinear.

In the case where the п¬Ѓgure 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: DesarguesвЂ™ Theorem

The case where the п¬Ѓgure is contained in a plane can be deduced from the
вЂњgeneralвЂќ 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 DesarguesвЂ™ 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 conп¬Ѓguration C
is not coplanar; so suppose it is. Let od be a line not in this plane, with d H, and DВў
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 DesarguesвЂ™ Theorem.
(If we are only given a plane initially, the crucial fact is that the plane can be
embedded in a 3-dimensional space.)

В G G
Remark The case where F 2 is not covered by this argument вЂ” can you see
why? вЂ” and, indeed, the projective plane over GFВҐ 2ВЁ contains no non-degenerate
Desargues conп¬Ѓguration: it only contains seven points! Nevertheless, DesarguesвЂ™
Theorem holds, in the sense that any meaningful degeneration of it is true in the
projective plane over GFВҐ 2ВЁ . We will not make an exception of this case.
It is also possible to prove DesarguesвЂ™ 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 п¬Ѓeld F . It has the same rank as V if this is п¬Ѓnite. Thus
\$
we have projective spaces PG n F and PGВҐ n F , standing in a dual relation to
ВЁВҐ H\$ 
ВЁ
one another. More precisely, we have a bijection between the п¬‚ats of PG n F ВЁВҐ
and those of PGВҐ n F , given by
ВЁ\$ 
( В  PВЁ Вў В¦В§ВҐ ' Вў ВЎ В  ВЁ
U AnnВҐ U f V: u U fu 0ВЁ
I ВҐ

This correspondence preserves incidence and reverses inclusion:
U1 U2 AnnВҐ U2 AnnВҐ U1
0 RВЁ
0 ВЁ
Q
В ВЁ
AnnВҐ U1 U2 AnnВҐ U1 AnnВҐ U2SВЁ
)
ВЁ

В ВЁ
AnnВҐ U1 U2 AnnВҐ U1 AnnВҐ U2
) ВЁ
HВЁ

Moreover, the (geometric) dimension of AnnВҐ U is n 1 dimВҐ U .  ВЁ ВЁ
This gives rise to a duality principle, where any conп¬Ѓguration theorem in pro-
jective space translates into another (over the opposite п¬Ѓeld) in which inclusions
are reversed and dimensions suitably modiп¬Ѓed. 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 brieп¬‚y to afп¬Ѓne 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 afп¬Ѓne space AGВҐ n F are the cosets of the vector subspaces of
ВЁ
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 afп¬Ѓne space is вЂњa vector space with no distinguished originвЂќ.
The other description is: AGВҐ n F is obtained from PGВҐ n F by deleting a
ВЁ ВЁ
hyperplane together with all the subspaces it contains.
The two descriptions are matched up as follows. Take the vector space
ВЁ  &&   ВҐ ВЎ В
В  F nT 1
V x x1 xn : x0 xn F
  Вў  && 

0
В
Let W be the hyperplane deп¬Ѓned by the equation x0 0. The points remaining
DВ
are rank 1 subspaces spanned by vectors with x0 0; each point has a unique
В
spanning vector with x0 1. Then the correspondence between points in the two
descriptions is given by U
1 x1 xn x xn
HВЁ  &&  ВҐ В¤&ВЁ  &&   ВҐ
I V 
  1
1.2. Projective spaces 7

(See Exercise 2.)
In AGВҐ n F , we say that two subspaces are parallel if (in the п¬Ѓrst description)
ВЁ
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 afп¬Ѓne space
as follows. To each parallel class of d-dimensional subspaces of AGВҐ n F cor-ВЁ
responds a unique d 1ВЁ -dimensional subspace of PGВҐ n 1 F . Adjoin to the
ВҐ ВЁ 
afп¬Ѓne space the points (and subspaces) of PGВҐ n 1 F , and adjoin to all mem-
ВЁ 
bers of a parallel class all the points in the corresponding subspace. The result is
PGВҐ n F .
ВЁ
The distinguished hyperplane is called the hyperplane at inп¬Ѓnity or ideal hy-
perplane. Thus, an afп¬Ѓne space can also be regarded as вЂњa projective space with a
distinguished hyperplaneвЂќ.
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 вЂњpoints at inп¬ЃnityвЂќ. Thus, the physical picture plane is an afп¬Ѓne 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 inп¬Ѓnity comprise a single line. Not that
the picture plane really is afп¬Ѓne rather than Euclidean; the ordinary distances in it
do not correspond to distances in the real world.

Exercises
1. Prove DesarguesвЂ™ Theorem in coordinates.
2. Show that the correspondence deп¬Ѓned in the text between the two descrip-
tions of afп¬Ѓne 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 inп¬Ѓnity 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 п¬Ѓgure can be
8 1. Projective spaces

drawn with the above restriction on the slopes.
(b) Investigate similarly how to draw a Desargues conп¬Ѓguration.

1.3 The вЂњFundamental Theorem of Projective GeometryвЂќ

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 PGВҐ n F to
ВЁВҐ ВЁ
itself. The theorem of the title of this section has two consequences: п¬Ѓrst, that iso-
morphic projective spaces have the same dimension and the same coordinatising
п¬Ѓeld; 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 GLВҐ n 1 F is the group of all non-singular linear
ВЁ 
В
transformations of V F nT 1 ; it is isomorphic to the group of invertible n 1ВЁ
X ВҐ
%
n 1ВЁ matrices over F. (In general, the determinant is not well-deп¬Ѓned, so we
ВҐ
cannot identify the invertible matrices with those having non-zero determinant.)
Any element of GLВҐ n 1 F maps subspaces of V into subspaces of the same
ВЁ 
rank, and preserves inclusion; so it induces a collineation of PGВҐ n F . The group
ВЁ
AutВҐ F of automorphisms of F has a coordinate-wise action on V nT 1 ; these trans-
ВЁ
formations also induce collineations. The group generated by GLВҐ n 1 F and ВЁ 
AutВҐ F (which is actually their semi-direct product) is denoted by О“LВҐ n 1 F ;
ВЁ ВЁ 
its elements are called semilinear transformations. The groups of collineations of
PGВҐ n F induced by GLВҐ n 1 F and О“LВҐ n 1 F are denoted by PGLВҐ n 1 F
ВЁ ВЁ  ВЁ  ВЁ 
and PО“LВҐ n 1 F , respectively.
ВЁ 
More generally, a semi-linear transformation from one vector space to another
is the composition of a linear transformation and a coordinate-wise п¬Ѓeld 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 вЂњFundamental Theorem of Projective GeometryвЂќ 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 п¬Ѓelds.

Corollary 1.7 (a) For n 1, the collineation group of PGВҐ n F is the group
W ВЁ
PО“LВҐ n 1 F .
ВЁ 
(b) The kernel of the action of О“LВҐ n 1 F on PGВҐ n F is the group of non-zero
ВЁ  ВЁ
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 п¬Ѓeld up to
isomorphism.

Proof We show п¬Ѓrst 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 sufп¬Ѓces to show that a semi-linear transformation which
п¬Ѓxes every point of PGВҐ n F is a scalar multiplication. So let v vПѓ A п¬Ѓx every
ВЁ 
point of PG n F , where Пѓ AutВҐ F and A GLВҐ n 1 F . Then every vector
ВЁВҐ Вў ВЁ Вў
ВЁ 
is mapped to a scalar multiple of itself. Let e0 en be the standard basis for V .
 Y& 

Then (since Пѓ п¬Ѓxes the standard basis vectors) we have ei A О»i ei for i 0 В   &&  В
n. 
Also,

О»0 e0 О»n en
В
e en A
 &&  ВҐ
 ВЁ  && 

0
О»ВҐ e0
В  en say,
 Y& 
 ВЁ
so О»0 О»n О».
В  &Y В  В

О» ВµПѓ О» 0
Now, for any Вµ F, the vector 1 Вµ0 0ВЁ is mapped to the vector 0ВЁ ;
ВўВ   &&    ВҐ
 ВҐ  &Y  

so we have О»Вµ ВµПѓ О». Thus
vПѓ A vПѓ О» О»v
В  В
10 1. Projective spaces

for any vector v, as required.
Note that the п¬Ѓeld automorphism Пѓ is conjugation by the element О» (that is,
ВµПѓ О»ВµО»` 1 ); in other words, an inner automorphism.
В
Now we prove that any isomorphism is semilinear. The strategy is similar.
Call an n 2ВЁ tuple of points special if no n 1 of them are linearly dependent.
ВҐ 
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
п¬Ѓeld).
(For, given a special tuple in the п¬Ѓrst space, spanning vectors for the п¬Ѓrst 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
 && 
  && 

п¬Ѓrst basis to the second also carries the п¬Ѓrst special tuple to the second.)
Let Оё be any isomorphism. Then there is a linear map П† which mimics the
effect of Оё on a special n 2ВЁ -tuple. Composing Оё with the inverse of П†, we
ВҐ
obtain an automorphism of PGВҐ n F which п¬Ѓxes the n 2ВЁ -tuple pointwise. We
ВЁ ВҐ
have to show that such an automorphism is the product of a scalar and a п¬Ѓeld
automorphism. (Note that, as we saw above, left and right multiplications by О»
differ by an inner automorphism.)
В
We assume that n 2; this simpliп¬Ѓes the argument, while retaining its essential
features. So let g be a collineation п¬Ѓxing the spans of e0 e1 e2 and e0 e1 e2 . We
 

use homogeneous coordinates, writing these vectors as 1 0 0ВЁ , 0 1 0ВЁ , 0 0 1ВЁ ,
ВҐ ВҐ ВҐ
and 1 1 1ВЁ , and denote the general point by x y zВЁ .
ВҐ ВҐ
aВЁ   ВҐ ВЎ
The points on the line 0 0 x2 , apart from
x 1 0 0ВЁ , have the form x 0 1ВЁ
 ВҐ ВҐ
for x F, and so can be identiп¬Ѓed with elements of F. Now the bijection be-
Вў
ВҐВЎ
tween this set and the set of points 0 y 1ВЁ on the line 0 x1 x2 , given by
ВҐ (ВЁ  

x 0 1ВЁ 0 x 1ВЁ , can be geometrically deп¬Ѓned in a way which is invariant un-
  ВҐ b   ВҐ
der collineations п¬Ѓxing the four reference points (see Fig. 1.3). The п¬Ѓgure also
shows that the coordinates of all points in the plane are determined.
Furthermore, the operations of addition and multiplication in F can be deп¬Ѓned
geometrically in the same sense (see Figures 1.4 and 1.6). (The deп¬Ѓnitions look
(   ВҐВЎ
more familiar if we take the line 1 x2 0ВЁ to be at inп¬Ѓnity, and draw the п¬Ѓgure
x

in the afп¬Ѓne plane with lines through 1 0 0ВЁ and 0 1 0ВЁ horizontal and vertical
ВҐ ВҐ
respectively. this has been done for addition in Figure 1.5; the reader should draw
1.3. The вЂњFundamental Theorem of Projective GeometryвЂќ 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
п¬Ѓxing our four basic points induces an automorphism of the п¬Ѓeld F, and its actions
on the coordinates agree. The theorem is proved.

A group G acting on a set в„¦ is said to be t-transitive if, given any two t-tuples
О±t and ОІ 1 ОІt of distinct elements of в„¦, some element of G carries
ВЁ  &Y  ВҐ
 ВЁ  &Y  ВҐ

О±1
the п¬Ѓrst 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 в„¦ agree everywhere.)
Since any two distinct points of PGВҐ n F are linearly independent, we see that
ВЁ
PО“LВҐ n 1 F (or even PGLВҐ n 1 F ) is 2-transitive on the points of PGВҐ n F . It
ВЁ  ВЁ  ВЁ
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

I will digress here to describe the analogous situation for PGВҐ 1 F , even ВЁ
though the FTPG does not apply in this case.

Proposition 1.8 (a) The group PGLВҐ 2 F is 3-transitive on the points of PGВҐ 1 F ,
ВЁ ВЁ
and is sharply 3-transitive if and only if F is commutative.

(b) There exist skew п¬Ѓelds F for which the group PGL 2 F is 4-transitive on ВЁВҐ
PGВҐ 1 F .
ВЁ
Proof The п¬Ѓrst part follows just as in the proof of the FTPG, since any three
points of PGВҐ 1 F have the property that no two are linearly dependent. Again,
ВЁ
as in that theorem, the stabiliser of the three points with coordinates 1 0ВЁ , 0 1ВЁ ВҐ ВҐ
and 1 1ВЁ is the group of inner automorphisms of F, and so is trivial if and only if
ВҐ
F is commutative.
1.3. The вЂњFundamental Theorem of Projective GeometryвЂќ 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: Afп¬Ѓne addition

There exist skew п¬Ѓelds 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 ; it is established by
a construction analogous to that of Higman, Neumann and Neumann  for
groups. Higman et al. used their construction to show that there exist groups
in which all non-identity elements are conjugate; CohnвЂ™s work shows that there
are multiplicative groups of skew п¬Ѓelds with this property. Note that such a п¬Ѓeld
DВ
has characteristic 2. For, if not, then 1 1 0, and any automorphism must п¬Ѓx

1 1.)

Finally, we consider collineations of afп¬Ѓne spaces.
Parallelism in an afп¬Ѓne space has an intrinsic, geometric deп¬Ѓnition. For two d-
п¬‚ats are parallel if and only if they are disjoint and some d 1ВЁ -п¬‚at contains both. ВҐ
It follows that any collineation of AGВҐ n F preserves parallelism. The hyperplane
ВЁ
at inп¬Ѓnity can be constructed from the parallel classes (as we saw in Section 1.2);
so any collineation of AGВҐ n F induces a collineation of this hyperplane, and
ВЁ
hence of the embedding PGВҐ n F . Hence:
ВЁ
Theorem 1.9 The collineation group of AGВҐ n F is the stabiliser of a hyperplane
ВЁ
in the collineation group of PGВҐ n F . ВЁ
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 AGВҐ n F is the semi-direct
W ВЁ
product of the additive group of F n and О“LВҐ n F . ВЁ
This group is denoted by AО“LВҐ n F . The additive group acts by translation,
ВЁ
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 deп¬Ѓnitions of AGВҐ n F given in ВЁ
the last section to deduce Proposition 1.10 from Theorem 1.9.

1.4 Finite projective spaces
Over the п¬Ѓnite п¬Ѓeld GFВҐ qВЁ , the n-dimensional projective and afп¬Ѓne spaces
and their collineation groups are п¬Ѓnite, and can be counted. In this section, we
display some of the relevant formulГ¦. We abbreviate PGВҐ n GFВҐ qВЁ to PGВҐ n qВЁ , &
ВЁ
 
and similarly for afп¬Ѓne spaces, collineation groups, etc.
A vector space of rank n over GFВҐ qВЁ is isomorphic to GFВҐ qВЁ 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 GFВҐ qВЁ is
q n 1ВЁ qn qВЁ qn qk 1
 yY&S  ВҐx  ВҐ
q k 1ВЁ qk qВЁ qk qk 1
 y&&S  ВҐx  ВҐ
n
Remark This number is called a Gaussian coefп¬Ѓcient, and is denoted by kВЃ q .
ВЂ
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 GLВҐ n qВЁ is 
n
1ВЁ qn qn qn 1
q qВЁ HВЁ `
 y&&S  ВҐx  ВҐ
The order of О“LВҐ n qВЁ is the above number multiplied by d, where q pd with p В

prime; and the orders of PGLВҐ n qВЁ and PО“LВҐ n qВЁ are obtained by dividing these
 
numbers by q 1ВЁ .
ВҐ
Proof An element of GLВҐ n qВЁ is uniquely determined by the image of the stan-

dard basis, which is an arbitrary basis of GFВҐ qВЁ 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 GFВҐ qВЁ has q 1 non-zero 
scalars, and its automorphism group has order d.

The formula for the Gaussian coefп¬Ѓcient 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 lвЂ™HЛ† pitalвЂ™s rule,
o
W
В
limqВ‚ 1 q a 1ВЁ qb 1ВЁ aВѓ b; it follows that
 В„&  ВҐ
ВҐВѓ
В… n n
В€В
В‡
lim 
kВ† kВ‰
qВ‚ 1 q
16 1. Projective spaces

This illustrates just one of the many ways in which subspaces of п¬Ѓnite vector
spaces resemble subsets of sets.
It follows immediately from Propopsition 1.11 that the numbers of k-dimensional
п¬‚ats in PGВҐ n qВЁ and AGВҐ n qВЁ are nT 1ВЃ q and qn k n q respectively.
1
ВЂ`
ВЂ
 
kT kВЃ
Projective and afп¬Ѓne spaces provide important examples of designs, whose
parameters can be expressed in terms of the Gaussian coefп¬Ѓcients.
A t-design with parameters v k О»ВЁ , or t-ВҐ v k О»ВЁ design, consists of a set X
ВҐ 
of v points, and a collection of k-element subsets of X called blocks, with the
Вђ
property that any t distinct points of X are contained in exactly О» blocks. Designs
were п¬Ѓrst used by statisticians, such as R. A. Fisher, for experimental design (e.g.
to facilitate analysis of variance). The terms вЂњdesignвЂќ and вЂњblockвЂќ, and the letter
v (the initial letter of вЂњvarietyвЂќ), reп¬‚ect this origin.
Proposition 1.13 (a) The points and m-dimensional п¬‚ats in PGВҐ n qВЁ form a 2- 
design with parameters
В…В‘ В… В…
n 1 m 1 n1
В†  В†  
1 1 m 1В†
q q qВ’

(b) The points and m-dimensional п¬‚ats of AGВҐ n qВЁ form a 2-design with param- 
eters В‘
В…
n1
qn qm 
 
m 1В† qВ’ 
2, then it is a 3-design, with О»
В  `` ВЂ В  n2
If q m 2ВЃ 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

В
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.
 Вѓ
(b) In AGВҐ n qВЁ , to count subspaces containing two points, we may assume

(by translation) that one of the points is the origin. An afп¬Ѓne п¬‚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
В
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 PGВҐ n qВЁ or 
AGВҐ n qВЁ by a п¬‚at F of dimension d is PG n d 1 qВЁ . (The п¬‚ats of the quotient
  ВҐ

space are precisely the п¬‚ats of the original space containing F.) This assertion is
true over any п¬Ѓeld 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 п¬‚ats in
В“ В“
PGВҐ n qВЁ or AGВҐ n qВЁ is PО“L n 1 qВЁ or AО“LВҐ n 1 qВЁ respectively, except in the
  В ВҐ 
 В 
afп¬Ѓne case with q 2 and m 1.

Proof By the results of Section 1.3, it sufп¬Ѓces to show that the entire geometry
can be recovered from the points and m-dimensional п¬‚ats. This follows immedi-
ately from two observations:

(a) the unique line containing two points is the intersection of all the m-dimensional
п¬‚ats containing them;

(b) except for afп¬Ѓne spaces over GFВҐ 2ВЁ , a set of points is a п¬‚at if and only if it
contains the line through any two of its points.

Afп¬Ѓne spaces over GFВҐ 2ВЁ 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 п¬‚ats of dimen-
sion m which contain them (for m 1); W
(bb) a set of points is a п¬‚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 п¬Ѓnite п¬Ѓeld
%
GFВҐ qВЁ is non-singular tends to a limit cВҐ qВЁ as n в€ћ, where 0 cВҐ qВЁ 1.
В“ В“R
18 1. Projective spaces

3. Prove that the total number F n of subspaces of a vector space of rank n
ВЁВҐ
over a given п¬Ѓnite п¬Ѓeld GFВҐ qВЁ satisп¬Ѓes the recurrence
В  n
Fn 1ВЁ 2F n q 1ВЁ F n 1ВЁ
ВҐ  ВҐ ВЁ ВҐ H ВҐ

4. Let S be an вЂњalphabetвЂќ of size q, with two distinguished elements 0 and 1
(but not necessarily a п¬Ѓnite п¬Ѓeld). A k n matrix with entries from S is (as usual)
%
in reduced echelon form if
В” it has no zero rows;
В” the п¬Ѓrst non-zero entry in any row is a 1;
В” the вЂњleading 1sвЂќ in later rows occur further to the right;
В” the other entries in the column of a вЂњleading 1вЂќ are all 0.
n
Prove that the number of k n matrices in reduced echelon form is kВЃ q . Verify in
% ВЂ
В  В
detail in the case n 4, k 2.
5. Use the result of Exercise 4 to prove the recurrence relation
В… В… В…
n n 1 n 1
В  qn 
kВ† k k 1В†
В†

q q q
2

Projective planes

Projective and afп¬Ѓne 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
rЛ† 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 п¬Ѓeld F, the geometry
В
PG 2ВЎ F has the following properties:
Вў
(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 satisп¬Ѓes conditions (PP1)-(PP3) above.
In a projective plane, let p and L be a point and line which are not incident.
The incidence deп¬Ѓnes 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 deп¬Ѓned to
В
be one less than this number. The order of PG 2ВЎ F is equal to the cardinality of
Вў В
F. (We saw in the last section that a projective line over GF qВў has 2 q q 1
ВЈ В¦
ВҐ
1В¤
В
points; so PG 2ВЎ qВў is a projective plane of order q. In the inп¬Ѓnite case, the claim
follows by simple cardinal arithmetic.)

19
20 2. Projective planes

Given a п¬Ѓnite projective plane of order n, each of the n 1 lines through a point
В¦
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
В¦В  В¦
points and lines form a 2- n2 n 1ВЎ n 1ВЎ 1Вў design. The converse is also true
В¦В¦ В¦
(see Exercise 2). В
Do there exist projective planes not of the form PG 2ВЎ F ? The easiest such
Вў
examples are inп¬Ѓnite; I give two completely different ones below. Finite examples
will appear later.

Example 1: Free planes. Start with any conп¬Ѓguration 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 doesnвЂ™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 conп¬Ѓguration violating DesarguesвЂ™ Theorem (for example, the Desargues
conп¬Ѓguration with the line pqr вЂњbrokenвЂќ into separate lines pq, qr, rp), then the
В
resulting plane doesnвЂ™t satisfy DesarguesвЂ™ Theorem, and so is not a PG 2ВЎ F . Вў
Example 2: Moulton planes. Take the ordinary real afп¬Ѓne 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 inп¬Ѓnite) slope are unaffected. This is an afп¬Ѓne plane, and
has a unique completion to a projective plane (see later). The resulting projective
plane fails DesarguesвЂ™ theorem. To see this, draw a Desargues conп¬Ѓguration 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 п¬Ѓrst examples of п¬Ѓnite planes in which DesarguesвЂ™ Theorem fails were
constructed by Veblen and Wedderburn . Many others have been found since,
but all known examples have prime power order. The BruckвЂ“Ryser Theorem 
asserts that, if a projective plane of order n exists, where n 1 or 2 (mod 4), then
В§
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  showed by an extensive computation that
there is no projective plane of order 10. The other values mentioned are undecided.
An afп¬Ѓne 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 EuclidвЂ™s вЂњpar-
allel postulateвЂќ. It is called вЂњPlayfairвЂ™s AxiomвЂќ, although it was stated explicitly
by Proclus. В
Again it holds that AG 2ВЎ F is an afп¬Ѓne plane. More generally, if a line and all
Вў
its points are removed from a projective plane, the result is an afп¬Ѓne plane. (The
removed points and line are said to be вЂњat inп¬ЃnityвЂќ. Two lines are parallel if and
only if they contain the same point at inп¬Ѓnity.
Conversely, let an afп¬Ѓne plane be given, with point set and line set . It
follows from (AP2) that parallelism is an equivalence relation on . Let be the В© 
set of equivalence classes. For each line L , let L L Q , where Q is the

ВҐ
parallel class containing L. Then the structure with point set , and line set
ВЁ

L : L , is a projective plane. Choosing as the line at inп¬Ѓnity, we
 "! 
   В© 
recover the original afп¬Ѓne plane.
We will have more to say about afп¬Ѓne planes in Section 3.5.

Exercises
1. Show that a structure which satisп¬Ѓes (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.
В¦В¦В
2. Show that a 2- n2 n 1ВЎ n 1ВЎ 1Вў design (with n 1) is a projective plane
В¦ #
of order n.
3. Show that, in a п¬Ѓnite afп¬Ѓne 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;
В¦
\$ there are n2 points;
\$ there are n 1 parallel classes with n lines in each.
В¦
(The number n is the order of the afп¬Ѓne plane.)
4. (The Friendship Theorem.) In a п¬Ѓnite society, any two individuals have a
unique common friend. Prove that there exists someone who is everyone elseвЂ™s
friend. В  В© В
[Let X be the set of individuals, F xВў : x X , where F xВў is the set  
ВҐ
of friends of X . Prove that, in any counterexample to the theorem, X is a Вў
projective plane, of order n, say. В
Now let A be the real matrix of order n2 n 1, with xВЎ yВў entry 1 if x and yВ¦В¦
are friends, 0 otherwise. Prove that

A2 nI JВЎ
В¦
ВҐ
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
В¦ (&
'
the fact that A has trace 0, calculate the multiplicity of the eigenvalue n, and '
hence show that n 1.] ВҐ
5. Show that any Desargues conп¬Ѓguration in a free projective plane must lie
within the starting conп¬Ѓguration. [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
В
PG 2ВЎ F s in the last section by the failure of DesarguesвЂ™ Theorem.
Вў
В
Theorem 2.1 A projective plane is isomorphic to PG 2ВЎ F for some F if and only Вў
if it satisп¬Ѓes DesarguesвЂ™ Theorem.

I do not propose to give a detailed proof of this important result; but some
comments on the proof are in order. В
We saw in Section 1.3 that, in PG 2ВЎ F , the п¬Ѓeld operations (addition and mul-
Вў
tiplication) can be deп¬Ѓned 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 deп¬Ѓne 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 п¬Ѓeld axiom translates into a certain вЂњconп¬Ѓguration
В
theoremвЂќ, so that the plane is a PG 2ВЎ F if and only if all these вЂњconп¬Ѓguration
Вў
theoremsвЂќ hold. What is not obvious, and quite remarkable, is that all these вЂњcon-
п¬Ѓguration theoremsвЂќ follow from DesarguesвЂ™ Theorem.
Another method, more difп¬Ѓcult in principle but much easier in detail, exploits
the relation between DesarguesвЂ™ Theorem and collineations.
Let p be a point and L a line. A central collineation with centre p and axis
L is a collineation п¬Ѓxing 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 В
centre p and axis L form a group. The plane is said to be pВЎ LВў -transitive if this
group permutes transitively the set M pВЎ L M for any line M L on p (or,
0)
 1  2ВҐ
equivalently, the set of lines on q different from L and pq, where q p is a point 2ВҐ
of L).

o
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9
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9 @ @ A 8A 8
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5
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