A Course on Elation Quadrangles pot

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A Course on
Elation Quadrangles

Koen Thas
Author:
Koen Thas
Department of Mathematics
Ghent University
Krijgslaan 281, S25
9000 Ghent
Belgium
E-mail:
2010 Mathematics Subject Classification: 05-02, 20-02, 51-02; 05B25, 05E18, 20B25, 20D15, 20D20, 51B25,
51E12
Key words: Generalized quadrangle, elation group, Moufang condition, p-group
ISBN 978-3-03719-110-1
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and the detailed bibliographic data are available on the Internet at .
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2012 European Mathematical Society
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9 8 7 6 5 4 3 2 1
To Caroline
“Les Perspecteurs”
A sketch (85cm  130cm) of the French artist Abraham Bosse (1602–1676) dating
from 1648, demonstrating the projecting method of Girard Desargues.
Preface
“To every loving, gentle-hearted friend,
to whom the present rhyme is soon to go
so that I may their written answer know (…)”
Translated from
A ciascun’alma presa e gentil core, La Vita Nuova
Dante Alighieri, 1295
Local Moufang conditions
In two famous papers [16], [17], Fong and Seitz showed that all finite Moufang gen-
eralized polygons were classical or dual classical. In fact, they obtained this result in
group theoretical terms (classifying finite split BN-pairs), but Tits remarked the simple
geometrical translation. And of course, the converse was already well known. In a
search for a synthetic “elementary” proof of the Fong–Seitz result for the specific case
of generalized quadrangles (which is the central and most difficult part in [16], [17]),
Payne and J.A. Thas noticed that when one looks at the group generated by all root-
elations and dual root-elations which stabilize a given point of a Moufang quadrangle,
the group fixes all lines incident with that point, and acts sharply transitively on its op-
posite points. Let us call a point with this property an elation point, and a generalized
quadrangle with such a point an elation generalized quadrangle. Kantor noticed in the
early 1980s that, starting from a group with a suitable family of subgroups satisfying

certain properties, one can construct an elation quadrangle from this data in a natural
way, such that the group acts as an elation group. This process can be easily reversed,
so as to obtain such group theoretical data starting from any elation quadrangle. This
observation is the precise analogon of the fact that, in a Moufang projective plane, any
line is a translation line, and when one singles out the definition of translation line and
translation plane, one can also translate the situation in group theoretical terms to a
group with certain subgroups, etc. In that case, one obtains a group of order n
2
with a
family of n C 1 subgroups of order n, two by two trivially intersecting (in the infinite
case, one has to require that the product of any two of these subgroups equals the entire
group and that the subgroups cover the group). And conversely, starting from such
group theoretical data, one readily reconstructs a translation plane for which the group
acts as a translation group. The essential difference in this correspondence between
planes and quadrangles is that, in the planar case, the translation group necessarily is
abelian, and this is not so for elation groups of generalized quadrangles. In the planar
case, this property allows one to define a “kernel”, which is some skew field over which
viii Preface
the translation group naturally becomes a vector space. In quadrangular theory, one
has to assume that the elation group is abelian to obtain a similar notion of kernel, and
then again, the abelian elation group can be seen as a vector space. In fact, one also
assumes that the quadrangle is finite, since there are some nontrivial obstructions when
passing to the infinite case.
Planes

//
Translation planes

Quadrangles
//

Elation quadrangles
In this monograph, we will focus on general finite elation quadrangles, so without
the commutativity assumption on the group. In the commutative case a rich theory
is available, and we refer the reader to [59] and the references therein for the (many)
details. Another basic difference with the nonabelian case is that an abelian elation
group is unique (both for planes and quadrangles). That is, there can only be at most one
(“complete”, that is, transitive on the appropriate point set) abelian elation group for a
given line in a projective plane or point in a generalized quadrangle, and it necessarily
is elementary abelian (in the finite case). As we will see in the present notes, this fact
is not true for general elation quadrangles. We will encounter examples which admit
different (t-maximal) elation groups with respect to the same elation point, and they
even can be nonisomorphic. (As a by-product, we will construct the first infinite class
of translation nets with similar properties.) Also, in the planar case and the abelian
quadrangular case, any i-root and dual i-root involving the translation line or the elation
point is Moufang, and the unique t-maximal elation group is generated by the Moufang
elations. In general, such properties do not hold for elation quadrangles. We will
obtain the first examples of finite elation quadrangles for which not every (dual) i-root
involving the elation point is Moufang.
So we first have to handle these standard structural questions as a set up for the
theory.
After Kantor’s observations, many infinite classes of finite generalized quadrangles
were constructed as elation quadrangles, through the identification of “Kantor families”
in appropriate groups. Moreover, up to a combination of point-line duality and Payne
integration, every known finite generalized quadrangle is an elation quadrangle. This
observation lies at the origin of the need for a structural theory for elation quadrangles,
which appears to be lacking in the literature. In fact, most of the foundations can be
found in Chapters 8 and 9 of [46], and that’s about it.
In the present book, I hope to fill up this gap.
Preface ix
Further outline

Let me briefly outline the contents of this book besides what was already mentioned.
First of all, let me mention that the three basic references on generalizedquadrangles
are the monographs “Finite Generalized Quadrangles” [44], [46]; “Symmetry in Finite
Generalized Quadrangles” [68] and “Translation Generalized Quadrangles” [59] (on
elation quadrangles with an abelian elation group). These works will only have a small
overlap with the present notes.
I describe, in detail, the beautiful result of Frohardt which solved Kantor’s con-
jecture in the case when the number of points of the (elation) quadrangle is at most
the number of lines. The latter conjecture is the prime power conjecture for elation
quadrangles, and states that the parameters of a finite elation generalized quadrangle
are powers of the same prime. Along the same lines of Frohardt’s proof, I present a nice
proof of X. Chen (which was never published) of a conjecture of Payne on the parame-
ters of skew translation quadrangles (which are elation quadrangles such that any dual
i-root involving the elation point is Moufang with respect to the same dual root group).
The positivity of this conjecture was independently proven by Dirk Hachenberger (in
a more general setting), and his proof is also in these notes.
I will also formulate several new questions, often motivated by obtained results.
Once the theory on the standard structural questions is worked out, we concentrate on
more specific problems, such as a fundamental question posed by Norbert Knarr on
the aforementioned local Moufang conditions (motivated by the idea whether there are
other, more natural, definitions for the concept of elation quadrangle).
Another aim is to emphasize the role of special p-groups and Moufang conditions
as central aspects of elation quadrangle theory.
In many occasions slightly different proofs are given than those provided in the
literature. Also, about seventy exercises of (usually) an elementary character are for-
mulated in the text. Exercises which are somewhat less elementary have been indicated
with a superscript “
#
”; exercises which come with a superscript “
c

” ought to be even
more challenging.
Mental note. Throughout this work, almost always the generalized quadrangles (and
related objects) we consider are finite, even when this is not explicitly mentioned. When
this is not the case, the reader will be able to deduce this.
Finally
The notes presented here are partially based on several lectures I gave on elation quad-
rangles. In particular, I think of the lecture I presented at the conference “Finite
Geometries” in La Roche (2004, Belgium), and several talks at the “Buildings confer-
ences” in Würzburg, Darmstadt and Münster, Germany. Also, I lectured on this subject
x Preface
at the University of Colorado at Denver, USA. These talks were often an inspiration
for further research, as were the conversations with members of the audience, such as
Bill Kantor, Norbert Knarr, Stanley E. Payne and Markus Stroppel.
A first version of the manuscript was finished during a Research in Pairs stay at
the Mathematisches Forschungsinstitut Oberwolfach, together with Stefaan De Winter
and Ernie Shult, in April 2007. Revised versions were written during the summer of
2010 and the autumn of 2011. In 2010, the counter example of the conjecture stated
in [69] was found.
Finally (really)
I wish to thank one of the anonymous referees for providing an extremely detailed
list of suggestions, remarks and typos which really helped me to write up a better,
final, version of the manuscript. I am also extremely grateful to Manfred Karbe of the
EMS Publishing House for his exceptional good (and pleasant) help in the process of
publishing this work. Finally, during most of the writing, I was a postdoctoral fellow
of the Fund for Scientific Research (FWO) – Flanders.
Oberwolfach, April 2007, Ghent, December 2011 Koen Thas
Contents
Preface vii
1 Generalized quadrangles 1

1.1 Elementary combinatorial preliminaries 1
1.2 Some group theory 9
1.3 Finite projective geometry 12
1.4 Finite classical examples and their duals 13
2 The Moufang condition 16
2.1 Moufang quadrangles 16
2.2 Generators and relations 17
2.3 Coxeter groups 18
2.4 BN-pairs of rank 2 and quadrangles 19
3 Elation quadrangles 24
3.1 Automorphisms of classical quadrangles 24
3.2 Elation generalized quadrangles 26
3.3 Maximality and completeness 27
3.4 Kantor families 27
3.5 The classical GQs as EGQs – second approach 29
4 Some features of special p-groups 31
4.1 The general Heisenberg group 31
4.2 Exact sequences and complexes 32
4.3 Group cohomology 35
4.4 Special and extra-special p-groups 37
4.5 Another approach 38
4.6 Lie algebras 39
4.7 Lie algebras from p-groups 41
5 Parameters of elation quadrangles and structure of elation groups 44
5.1 Parameters of elation quadrangles 44
5.2 Skew translation quadrangles 46
5.3 F -Factors 47
5.4 Parameters of STGQs 49
xii Contents
6 Standard elations and flock quadrangles 50

6.1 Flock quadrangles 50
6.2 Fundamental theorem of q-clan geometry 52
6.3 A special elation 55
6.4 The nitty gritty 55
6.5 A special elation, once again 57
6.6 Standard elations in flock GQs 58
6.7 The general case 62
7 Foundations of EGQs 64
7.1 An application of Burnside’s lemma 64
7.2 Implications 66
7.3 Intermezzo – SPGQs 67
7.4 The classical and dual classical examples 68
7.5 Elation groups for flock GQs and their duals 69
7.6 Dual TGQs which are also EGQs 70
7.7 GQs of order .k  1; k C1/ and their duals 76
8 Elation quadrangles with nonisomorphic elation groups 78
8.1 A nonisomorphism criterion 78
8.2 An example: H.3; q
2
/, q even 81
8.3 Group and GQ automorphisms 81
8.4 Appendix: GQs not having property ./ 82
9 Application: Existence of translation nets 84
9.1 Translation nets 84
9.2 Construction 84
10 Elations of dual translation quadrangles 86
10.1 Main result 86
10.2 Payne’s question in a more general setting 88
10.3 Recent results 88
11 Local Moufang conditions 91

11.1 Formulation 91
11.2 Proof of the first main theorem 92
11.3 Solution of Knarr’s question 100
11.4 Appendix: GQs with a center of transitivity (and s Ä t) 100
Bibliography 105
Symbols 111
Index 113
1
Generalized quadrangles
We start this chapter by introducing some combinatorial and group theoretical notions.
We then proceed to define the prototypes of finite generalized quadrangles.
1.1 Elementary combinatorial preliminaries
We concisely review some basic notions taken from the theory of generalized quadran-
gles, for the sake of convenience.
1.1.1 Rank 2 geometries. A rank 2 geometry or point-line geometry is a triple  D
.P ; B; I/, for which P and B are disjoint (nonempty) sets of objects called points and
lines respectively, and for which I is a symmetric point-line relationcalled an “incidence
relation”; so I Â .P  B/ [ .B  P / and .x; L/ 2 I if and only if .L; x/ 2 I. If
.x; L/ 2 I, we also write x I L or L I x.If.x; L/ … I, we write x I

L or L I

x.
1.1.2 Generalized quadrangles. A generalized quadrangle (GQ) of order .s; t/ is a
point-line incidence geometry à D .P ; B; I/ satisfying the following axioms:
(i) each point is incident with t C1 lines (t  1) and two distinct points are incident
with at most one line;
(ii) each line is incident with s C 1 points (s  1);
(iii) if p is a point and L is a line not incident with p, then there is a unique point-line
pair .q; M / such that p I M I q I L.

In this definition, s and t are allowed to be infinite cardinals.
Exercise. Let à D .P ; B; I/ be a GQ of order .s; t/ with s; t 2 N. Show that
jP jD.s C 1/.st C1/ and jBjD.t C 1/.st C1/.
This exercise shows that if s and
t are finite, then jP jand jBj also are. In that case,
we call the quadrangle finite.Ifs; t > 1, Ã is thick; if one of s, t equals 1, Ã is thin.A
thin GQ of order .s; 1/ is also called a grid, while a thin GQ of order .1; t/ is a dual
grid. A GQ of order .1; 1/ is both a grid and a dual grid – it is an ordinary quadrangle.
If s D t, then à is also said to be of order s.
There is a parameter-free way to introduce generalized quadrangles, as follows.
A rank 2 geometry à D .P ; B; I/ is a thick generalized quadrangle if the following
axioms are satisfied:
2 1 Generalized quadrangles
Figure 1.1. A grid of order .3; 1/.
(a) there are no ordinary digons and triangles contained in Ã;
(b) every two elements of P [ B are contained in an ordinary quadrangle;
(c) there exists an ordinary pentagon.
In (a), (b), (c), ordinary digons, triangles, quadrangles and pentagons are meant to
be induced subgeometries.
Exercise. Show that if à satisfies (a)–(c), there exist constants s and t such that each
line is incident with s C 1 points, and each point is incident with t C 1 lines.
Exercise. Suppose that (a) and (b) are satisfied, but not (c). Show that à has a thin
structure, in the sense that either each point is incident with two lines, or each line is
incident with two points.
Suppose that .p; L/ … I. Then by proj
L
p we denote the unique point on L collinear
with p. Dually, proj
p
L is the unique line incident with p concurrent with L.

1.1.3 Duality. There is a point-line duality for GQs of order .s; t/ for which in any
definition or theorem the words “point” and “line” are interchanged and also the pa-
rameters. (If à D .P ; B; I/ is a GQ of order .s; t/, Ã
D
D .B; P ; I/ is a GQ of order
.t; s/.) A duality  from the GQ Ã to its dual Ã
D
is a map that bijectively sends points
of à to lines of Ã
D
, lines of à to points of Ã
D
, while preserving incidence. (This notion
will only be needed in a later chapter for formal reasons.)
Exercise. Show that there is a natural one-to-one correspondence between automor-
phisms of à (defined further in this section) and dualities from à to Ã
D
.
1.1 Elementary combinatorial preliminaries 3
−→
D
Figure 1.2. Duality. The left-hand side is a grid of order .3; 1/; the right-hand side its dual – a
GQ of order .1; 3/.
1.1.4 Inequalities of Higman. Let à D .P ; B; I/ be a finite thick GQ of order .s; t /.
Then we have t Ä s
2
and, dually, s Ä t
2
(“Inequality of Higman”); see [44], 1.2.3.
Also, by [44], 1.2.2, we have that

st.s C 1/.t C 1/ Á 0 mod s C t:
1.1.5 Collinearity and concurrency. Let p and q be (not necessarily distinct) points
of the GQ Ã; we write p  q and call these points collinear, provided that there is
some line L such that p I L I q. Dually, for L; M 2 B, we write L  M when L and
M are concurrent.
For p 2 P , put
p
?
Dfq 2 P j q  pg:
Note that p 2 p
?
.
uv
w 2fu; vg
?
Figure 1.3. The “perp” of fu; vg.
If two points are not collinear, we also say they are opposite. Same for lines. A
flag is an incident point-line pair.
4 1 Generalized quadrangles
1.1.6 Ovoids. An ovoid of a GQ is a set of points O such that each line contains
exactly one of its points.
Exercise. Let à be a GQ of order .s; t/. Show that the number of points of an ovoid
is st C 1.
1.1.7 Regularity. For a set of distinct points S (or lines), we denote
T
w2S
w
?
also
by S

?
. In particular, let S Dfp; qg be a set of two points; then jfp; qg
?
jDs C 1 or
t C 1, according as p  q or p ¦ q, respectively. A set such as fp; qg
?
is called a
trace; it is “trivial” when p  q ¤ p. For a set S such as above, we introduce S
??
as
S
??
D .S
?
/
?
:
For p ¤ q distinct points, we have that jfp; qg
??
jDs C1 or jfp; qg
??
jÄt C1
according as p  q or p ¦ q, respectively. If p  q, p ¤ q,orifp ¦ q and
jfp; qg
??
jDt C 1, we say that the pair fp; qg is regular. The point p is regular
provided fp; qgis regular for every q 2 P nfpg. Regularity for lines is defined dually.
Exercise. Prove that either s D 1 or t Ä s if à has a regular pair of noncollinear points
(see [44], 1.3.6).
A net of order k and degree r is a point-line incidence geometry N D .P ; B; I/

satisfying the following axioms:
(i) each point is incident with r lines (r  2) and two distinct points are incident
with at most one line;
(ii) each line is incident with k points (k  2);
(iii) if p is a point and L is a line not incident with p, then there is a unique line M
incident with p and not concurrent with L.
We say that .k; r/ “are” the parameters of the net. Sometimes we also speak of “.k; r/-
net”.
Exercise. Show that jP jDk
2
and jBjDkr.
Exercise. Show that a net N of degree r and order k is an affine plane of order n if
and only if r D k C1.
Theorem 1.1 ([44], 1.3.1). Let p be a regular point of a finite GQ Ã D .P ; B; I/ of
order .s; t/, s ¤ 1 ¤ t . Then the incidence structure with
• point set p
?
nfpg,
• with line set the set of spans fq; rg
??
, where q and r are noncollinear points of
p
?
nfpg,
and with the natural incidence, is the dual of a net of order s and degree t C 1.
1.1 Elementary combinatorial preliminaries 5
If in particular s D t, there arises a dual affine plane of order s. (Also, in the
case s D t, the incidence structure 
p
with point set p

?
, with line set the set of spans
fq; rg
??
, where q and r are different points in p
?
, and with the natural incidence, is
a projective plane of order s.)
Proof. We leave the proof as a straightforward exercise to the reader. 
Exercise. Come up with an “infinite version” of Theorem 1.1.
1.1.8 Antiregularity. The pair of points fx; yg, x ¦ y,isantiregular if jfx; yg
?
\
z
?
jÄ2 for all z 2 P nfx; yg. The point x is antiregular if fx; yg is antiregular for
each y 2 P n x
?
.
1.1.9 Triads. A triad of points of a GQ is a set of three pairwise noncollinear points.
Let fx;y; zgbe a triad of points in a thick finite GQ of order .s; s
2
/. Then jfx;y; zg
?
jD
s C 1; see [44], 1.2.4. Obviously, jfx; y;zg
??
jÄs C 1; if equality holds, the triad
fx; y; zg is called 3-regular. Furthermore, a point is 3-regular provided all triads of
which it is a member are 3-regular.

1.1.10 Automorphisms. An automorphism or collineation ofaGQÃ D .P ; B; I/ is
a permutation of P [ B which preserves P , B and I. The set of automorphisms of a
GQ Ã is a group, called the automorphism group of Ã, which is denoted by Aut.Ã/.A
whorl about a point x is just an automorphism fixing it linewise. A point x is a center
of transitivity provided that the group of whorls about x is transitive on the points of
P n x
?
.Anelation with center x is an automorphism of à which either is the trivial
automorphism, or it fixes x linewise and has no fixed points in P n x
?
.
Exercise. Show that a (not necessarily thin, nor finite) GQ Ã has the same automor-
phism group as its point-line dual.
1.1.11 Symmetry. A collineation  of Ã, a thick finite GQ of order .s; t/, that fixes all
lines meeting a fixed line L is called a symmetry about L. If the group of symmetries
about L has the maximum possible order, s, then L is called an axis of symmetry.
Dually, one speaks of a center of symmetry.
Exercise. Show that not only is a symmetry about L an elation about L, but that it is
also an elation about each point incident with L. (One is allowed to use Theorem 1.6
below.)
1.1.12 SubGQs. A subquadrangle, or also subGQ, Ã
0
D .P
0
; B
0
; I
0
/ ofaGQÃ D
.P ; B; I/ is a GQ for which P

0
 P , B
0
 B, and where I
0
is the restriction of I to
.P
0
 B
0
/ [ .B
0
 P
0
/. A subGQ Ã
0
of order .s; t
0
/ of a finite GQ Ã of order .s; t/ is
called full. Dually we define ideal subGQs.
6 1 Generalized quadrangles
The following results will sometimes be used without further reference. Theo-
rems 1.4, 1.5 and 1.6 can be obtained as easy exercises. For the proofs of Theorems 1.2
and 1.3, we refer to [44]. In all of the statements below up to Theorem 1.6, the gener-
alized quadrangles are supposed to be finite.
Theorem 1.2 ([44], 2.2.1). Let Ã
0
be a proper subquadrangle of order .s
0
;t

0
/ of the
GQ Ã of order .s; t /. Then either s D s
0
or s  s
0
t
0
.Ifs D s
0
, then each external
point of Ã
0
is collinear with the st
0
C 1 points of an ovoid of Ã
0
;ifs D s
0
t
0
, then each
external point of Ã
0
is collinear with exactly 1 Cs
0
points of Ã
0
.
Theorem 1.3 ([44], 2.2.2). Let Ã

0
be a proper subquadrangle of the GQ Ã, where Ã
has order .s; t/ and Ã
0
has order .s; t
0
/ (so t>t
0
). Then the following hold.
(1) t  s;ifs D t, then t
0
D 1.
(2) If s>1, then t
0
Ä s;ift
0
D s  2, then t D s
2
.
(3) If s D 1, then 1 Ä t
0
<tis the only restriction on t
0
.
(4) If s>1and t
0
>1, then
p
s Ä t
0

Ä s and s
3=2
Ä t Ä s
2
.
(5) If t D s
3=2
>1and t
0
>1, then t
0
D
p
s.
(6) Let Ã
0
have a proper subquadrangle Ã
00
of order .s; t
00
/, s>1. Then t
00
D 1,
t
0
D s and t D s
2
.
Theorem 1.4 ([44], 2.3.1). Let Ã
0

D .P
0
; B
0
; I
0
/ be a substructure of the GQ Ã of
order .s; t/ so that the following two conditions are satisfied:
(i) if x; y 2 P
0
are distinct points of Ã
0
and L is a line of à such that x I L I y, then
L 2 B
0
;
(ii) each element of B
0
is incident with s C 1 elements of P
0
.
Then there are four possibilities:
(1) Ã
0
is a dual grid, so s D 1;
(2) the elements of B
0
are lines which are incident with a distinguished point of P ,
and P
0

consists of those points of P which are incident with these lines;
(3) B
0
D;and P
0
is a set of pairwise noncollinear points of P ;
(4) Ã
0
is a subquadrangle of order .s; t
0
/.
The following result is now easy to prove.
Theorem 1.5 ([44], 2.4.1). Let  be an automorphism of the GQ à D .P ; B; I/ of
order .s; t/. The substructure Ã
Â
D .P
Â
; B
Â
; I
Â
/ of à which consists of the fixed
elements of  must be given by (at least) one of the following:
(i) B
Â
D;and P
Â
is a set of pairwise noncollinear points;
(i
0

) P
Â
D;and B
Â
is a set of pairwise nonconcurrent lines;
1.1 Elementary combinatorial preliminaries 7
(ii) P
Â
contains a point x so that y  x for each y 2 P
Â
, and each line of B
Â
is
incident with x;
(ii
0
) B
Â
contains a line L so that M  L for each M 2 B
Â
, and each point of P
Â
is
incident with L;
(iii) Ã
Â
is a grid;
(iii
0
) Ã

Â
is a dual grid;
(iv) Ã
Â
is a subGQ of à of order .s
0
;t
0
/, s
0
;t
0
 2.
Finally, we recall a result on fixed elements structures of whorls.
Theorem 1.6 ([44], 8.1.1). Let  be a nontrivial whorl aboutp of the GQ à D .P ; B; I/
of order .s; t/, s ¤ 1 ¤ t . Then one of the following must hold for the fixed element
structure Ã
Â
D .P
Â
; B
Â
; I
Â
/.
(1) y
Â
¤ y for each y 2 P np
?
.

(2) There is a point y, y ¦ p, for which y
Â
D y. Put V Dfp; yg
?
and U D V
?
.
Then V [fp; ygÂP
Â
 V [ U , and L 2 B
Â
if and only if L joins a point of
V with a point of U \P
Â
.
(3) Ã
Â
is a subGQ of order .s
0
;t/, where 2 Ä s
0
Ä s=t Ä t , and hence t<s.
Exercise. Create an “infinite version” of Theorem 1.6.
1.1.13 Nets and subquadrangles. The following theorem is taken from [64] and
implies that a net which arises from a regular point in a thick finite GQ as earlier
explained cannot contain proper subnets of the same degree and different from an
affine plane.
Theorem 1.7 ([64]). Suppose that à D .P ; B; I/ isaGQoforder.s; t/, s; t ¤ 1, with
a regular point p. Let N
p

be the net which arises from p, and suppose N
0
p
is a subnet
of the same degree as N
p
. Then we have the following possibilities:
(1) N
0
p
coincides with N
p
;
(2) N
0
p
is an affine plane of order t and s D t
2
; also, from N
0
p
there arises a proper
subquadrangle of à of order t having p as a regular point.
If, conversely, Ã has a proper subquadrangle containing the point p and of order
.s
0
;t/with s
0
¤ 1, then it is of order t, and hence s D t
2

. Also, there arises a proper
subnet of N
p
which is an affine plane of order t.
Proof. First suppose that à contains aproper subquadrangle Ã
0
of order .s
0
;t/, s
0
;t ¤ 1,
containing the point p. Then p is also regular in Ã
0
and since s
0
¤ 1, it follows that
s
0
 t. By Theorem 1.3 this implies that s
0
D t and that s D t
2
. By Theorem 1.1, the
net N
0
p
arising from the point p in Ã
0
is an affine plane of order t, and this net is clearly
a subnet of the net which arises from the point p in Ã.

8 1 Generalized quadrangles
Conversely, suppose that N
p
is the net which arises from the regular point p in the
GQ Ã, and that it contains a proper subnet N
0
p
of the same degree. In the following,
we identify points of the net with the corresponding spans of points in the GQ, and we
use the same notation.
Suppose that P
1
;P
2
;:::;P
k
are the points of N
0
p
, define a point set P
0
of à as
consisting of the points of Œ
S
P
i
 [ Œ
S
P
?

i
, and define B
0
as the set of all lines of Ã
through a point of P
0
. Then it is not hard to check that the following properties are
satisfied for the geometry Ã
0
D .P
0
; B
0
; I
0
/, with I
0
D I \Œ.P
0
 B
0
/ [ .B
0
 P
0
/:
(1) any point of P
0
is incident with t C 1 lines of B
0

;
(2) if two lines of B
0
intersect in Ã, then they also intersect in Ã
0
.
Then by the dual of Theorem 1.4, Ã
0
is a proper subquadrangle of order .s
0
;t/,
s
0
¤ 1, and analogously as in the beginning of the proof, we have that s
0
D t and
s D t
2
. Also, the affine plane of order t which arises from the regular point p in this
subquadrangle is the subnet N
0
p
. 
Corollary 1.8 ([64]). A net N which is attached to a regular point of a GQ contains
no proper subnet of the same degree as N , other than ( possibly) an affine plane.
Corollary 1.9 ([64]). Suppose that p is a regular point of the GQ Ã of order .s; t/,
s; t ¤ 1, and let N
p
be the corresponding net. If s ¤ t
2

, then N
p
contains no proper
subnet of degree t C 1.
The following corollary tells us that nets which arise from a regular point of a GQ
and which do not contain affine planes are very “irregular”.
Corollary 1.10 ([64]). Let p be a regular point of a GQ Ã of order .s; t/, s; t ¤ 1, and
suppose that N
p
is the corresponding net. Moreover, suppose that s ¤ t
2
.Ifu, v and
w are distinct lines of N
p
for which w ¦ u  v, then these lines generate (under the
taking of GQ spans) the whole net.
Proof. Consider the points of p
?
nfpg which correspond to the lines u; v; w of N
p
,
and denote them respectively in the same way. Then by Theorem 1.4, u, v and w
generate a (not necessarily proper) subGQ Ã
0
of à of order .s
0
;t/, where s
0
>1.By
Theorem 1.7 this implies that Ã

0
D Ã, since s ¤ t
2
. Hence u, v and w generate N
p
.

Lemma 1.11. Suppose that à isaGQoforder.s; s
2
/, s ¤ 1, and suppose that Ã
0
and Ã
00
are two proper subquadrangles of à of order s. Then one of the following
possibilities occurs:
(1) Ã
0
\Ã
00
is a set of s
2
C1 pairwise noncollinear points (i.e., an ovoid) of Ã
0
and
Ã
00
;
(2) Ã
0
\ Ã

00
consists of a point p of Ã
0
(and Ã
00
), together with all lines of Ã
0
(and
Ã
00
) through this point, and all points of Ã
0
(and Ã
00
) incident with these lines;
1.2 Some group theory 9
(3) Ã
0
\ Ã
00
isaGQoforder.s; 1/;
(4) Ã
0
D Ã
00
.
Exercise. Prove Lemma 1.11. Note that every line of a thick GQ of order .s; s
2
/
intersects any subGQ of order s. Then use Theorem 1.4 and a simple counting argument.

Theorem 1.12 ([64]). Suppose that à is a generalized quadrangle of order .s; t/,
s; t ¤ 1, and suppose that  is a nontrivial whorl about a regular point p. Also,
suppose that  fixes distinct points q; r and u of p
?
nfpgfor which q  r and q ¦ u.
Then we have one of the following possibilities.
(1) We have that s D t
2
and à contains a proper subquadrangle Ã
0
of order t.
Moreover, if  is not an elation, then Ã
0
is fixed pointwise by .
(2)  is a nontrivial symmetry about p.
Proof. It is clear that if v and w are noncollinear points of p
?
which are fixed by
a whorl about p, then every point of the span fv; wg
??
is also fixed by the whorl.
Now suppose that N
p
is the net which arises from p, and suppose that N
0
p
is the (not
necessarily proper) subnet of N
p
of degree t C 1 which is generated by u, q and r.

Then every point of N
0
p
is fixed by  by the previous observation. If N
0
p
is proper,
then by Theorem 1.7 it is an affine plane of order t and s D t
2
. Also, there arises a
proper subquadrangle Ã
0
of à of order t.If is not an elation, then by Theorem 1.6 it
follows that there is a proper subquadrangle Ã

of order .s
0
;t/, s
0
¤ 1, which is fixed
pointwise (and then also linewise) by . Since Ã

has a regular point, we have that
s
0
 t . By Theorem 1.3, Ã
0
is necessarily of order t . From Lemma 1.11 now follows
that Ã


D Ã
0
.
If N
0
p
D N
p
, then every point of p
?
is fixed by . Since  is not the identity, it
follows from Theorem 1.6 that  is an elation and hence a symmetry about p. 
1.2 Some group theory
We review some basic notions of group theory.
1.2.1 Identity. We denote the identity element of a group often by id or 1; a group G
without its identity is denoted by G

.
1.2.2 Permutation groups. We usually denote a permutation group by .G; X/, where
G acts on X. We denote permutation action exponentially and let elements act on the
right, such that each element g of G defines a permutation g W X ! X of X and the
permutation defined by gh, g; h 2 G,isgivenby
ghW X ! X; x 7! .x
g
/
h
:
10 1 Generalized quadrangles
1.2.3 Commutators. Let G be a group, and let g; h 2 G. The conjugate of g by h is
g

h
D h
1
gh. The commutator of g and h is equal to
Œg; h D g
1
h
1
gh:
Note that the commutator map
W G  G ! G; .g; h/ 7! Œg; h;
is not symmetrical; as Œg; h
1
D Œh; g, we have that Œg; h D Œh; g if and only if
Œg; h is an involution.
The commutator of two subsets A and B of a group G is the subgroup ŒA; B
generated by all elements Œa; b, with a 2 A and b 2 B. The commutator subgroup
of G is ŒG; G, or sometimes G
0
. Two subgroups A and B centralize each other if
ŒA; B Dfidg. The subgroup A normalizes B if B
a
D B for all a 2 A, which is
equivalent with ŒA; B Ä B.
Inductively, we define the n-th central derivative L
nC1
.G/ D ŒG; G
Œn
of G as
ŒG; ŒG; G

Œn1
, and the n-th normal derivative ŒG; G
.n/
as ŒŒG; G
.n1/
;ŒG;G
.n1/
.
For n D 0, the 0-th central and normal derivative are by definition equal to G itself.
The series
L
1
.G/; L
2
.G/; : : :
is called the lower central series of G. If, for some natural number n, ŒG; G
.n/
Dfidg,
and ŒG; G
.n1/
¤fidg, then we say that G is solvable (soluble) of length n.If
ŒG; G
Œn
Dfidg and ŒG; G
Œn1
¤fidg, then we say that G is nilpotent of class n.A
group G is called perfect if G D ŒG; G D G
0
.
The center of a group is the set of elements that commute with every other element,

i.e., Z.G/ Dfz 2 G j Œz; g D id for all g 2 Gg. Clearly, if a group G is nilpotent of
class n, then the .n  1/-th central derivative is a nontrivial subgroup of Z.G/.
1.2.4 Central products. A group H is an internal central product of its subgroups
M and N if both N and M are normal subgroups of H for which N \ M Ä Z.H/
and NM D H . Now let M and N be two groups, N

Ä Z.N /, M

Ä Z.M/, and
 W N

! M

an isomorphism. Then the quotient Q D .N M /=K, where K is the normal subgroup
f.n; m/ j n 2 N

;m2 M

; Â.n/m D 1g, is the external central product of N and
M provided by the data .N

;M

;Â/.
Exercise. Work out the connection between internal and external central product of
groups.
(The terms “internal/external” will be dropped if it is clear which type of central
product is considered.)
1.2 Some group theory 11
1.2.5 p-Groups and Hall groups. For a prime number p,ap-group is a group of

order p
n
, for some natural number n ¤ 0.ASylow p-subgroup of a finite group G is
a p-subgroup of some order p
n
such that p
nC1
does not divide jGj. Let  be a set of
primes dividing jGj for a finite group G. Then a -subgroup is a subgroup of which
the set of prime divisors is .
The following result is basic.
Theorem 1.13 ([19], Chapter 1). A finite group is nilpotent if and only if it is the direct
product of its Sylow subgroups.
A Hall -subgroup of a finite group G, where  Â .G/, and .G/ is the set of
primes dividing jGj, is a subgroup of size
Q
p2
p
n
p
, where p
n
p
denotes the largest
power of p that divides jGj.
Theorem 1.14 (Hall’s Theorem, [19], Chapter 6). Let G be a finite solvable group and
 a set of primes. Then
(a) G possesses a Hall -subgroup;
(b) G acts transitively on its Hall -subgroups by conjugation;
(c) any -subgroup of G is contained in some Hall -subgroup.

Let p and q be primes. A pq-group is a group of order p
a
q
b
for some natural
numbers a and b. A classical result of Burnside states the following.
Theorem 1.15 ([19], Chapter 4). A pq-group is solvable.
Let R be a finite group. The Frattini group ˆ.R/ of R is the intersection of all
proper maximal subgroups, or is R if R has no such subgroups.
Exercise. Let P be a finite p-group. Show that ŒP; P P
p
D ˆ.P /, where P
p
D
hw
p
j w 2 P i.
1.2.6 Frobenius groups. Suppose that .G; X/ is a permutation group (where G acts
on X) which satisfies the following properties:
(1) G acts transitively but not sharply transitively on X;
(2) there is no nontrivial element of G with more than one fixed point in X.
Then .G; X/ is a Frobenius group (or G is a Frobenius group in its action on X ).
Define N Â G by
N Dfg 2 G j f.g/ D 0g[f1g;
where f.g/ is the number of fixed points of g in X. Then N is called the Frobenius
kernel of G (or of .G; X /), and we have the following well-known result.
Theorem 1.16 (Theorem of Frobenius, [19], Chapter 2). Suppose that jGj is finite.
The Frobenius kernel N is a normal regular subgroup of G. Moreover, jG
x
j divides

jN j1 for any x 2 X, and G D N Ì G
x
.
12 1 Generalized quadrangles
1.2.7 Simple groups. A group is simple if it does not contain any proper nontrivial
normal subgroups. A group G is almost simple if S Ä G Ä Aut.S/, with S a simple
group and Aut.S/ its automorphism group.
1.3 Finite projective geometry
1.3.1 Projective spaces. Below, F
q
denotes the finite field with q elements, q a prime
power. Let K be any field, and denote by V.n;K/ the n-dimensional vector space over
K, n a nonzero natural number. If K D F
q
is a finite field, we also use the notation
V.n;q/. Define the .n  1/-dimensional projective space PG.n  1; K/ over K as
the geometry of all subspaces of V.n;q/ ordered by set inclusion; more precisely,
it is V.n;K/ equipped with the equivalence relation “” of proportionality, with the
induced subspace structure. If K D F
q
is finite, we also use the notation PG.n 1; q/.
The projective space PG.1; K/ is the empty set, and has dimension 1. In general, if
W is a w-dimensional K-vector subspace of V.n;K/, it induces a .w 1/-dimensional
projective subspace of V.n;K/=DPG.n  1; K/.
Exercise. Let K D F
q
. Show that a d(-dimensional)-subspace of PG.n1; q/ contains
.q
d C1
1/=.q 1/ points. In particular, PG.n 1; q/ has .q

n
1/=.q 1/ points. It
also has .q
n
 1/=.q  1/ hyperplanes (= .n  2/-dimensional subspaces).
1.3.2 Collineation groups. An automorphism or collineation of a finite projective
space is an incidence and dimension (“type”) preservingbijection of the setof subspaces
to itself. It can be shown that any automorphism of a PG.n; q/, n 2 N and n  2, F
q
a finite field, necessarily has the following form:
 W x
T
! A.x

/
T
;
where A 2 GL
nC1
.q/,  is a field automorphism of F
q
, the homogeneous coordinate
x D .x
0
;x
1
;:::;x
n
/ represents a point of the space (which is determined up to a
scalar), and x


D .x

0
;x

1
;:::;x

n
/ (recall that x

i
is the image of x
i
under ).
Here, vectors are identified with row matrices without any further notice. The
set of automorphisms of a projective space naturally forms a group, and in case
of PG.n; q/, n  2, this group is denoted by PL
nC1
.q/. The normal subgroup
of PL
nC1
.q/ which consists of all automorphisms for which the companion field
automorphism  is the identity, is the projective general linear group, and denoted
by PGL
nC1
.q/. So PGL
nC1
.q/ D GL

nC1
.q/=Z.GL
nC1
.q//, where Z.GL
nC1
.q//
is the central subgroup of all scalar matrices of GL
nC1
.q/. Similarly one defines
PSL
nC1
.q/ D SL
nC1
.q/=Z.SL
nC1
.q//, where Z.SL
nC1
.q// is the central subgroup
of all scalar matrices of SL
nC1
.q/ with unit determinant.
An elation of PG.n; q/ is an automorphism of which the fixed points structure
precisely is a hyperplane (the “axis” of the elation), or the space itself. A homology
1.4 Finite classical examples and their duals 13
either is the identity, or it is an automorphism that fixes a hyperplane pointwise, and
one further point not contained in that hyperplane.
Exercise. Show that each nontrivial elation of PG.n; q/, n 2 N and n  2, has a
unique center, that is, a point which is fixed linewise (and necessarily contained in the
axis).
1.4 Finite classical examples and their duals

In this section we will introduce some classes of finite rank 2 geometries which are
known as the finite “classical generalized quadrangles”. (Tits was the first to iden-
tify them as generalized quadrangles – see Dembowski [15].) Their point-line duals
are called the dual classical generalized quadrangles. The classical quadrangles are
characterized by the fact that they are fully embedded in finite projective space –
see Chapter 4 in [44] for details. Recall that a full embedding of a rank 2 geometry
 D .P ; B; I/ in a projective space P , is an injection
ÃW P ,! P .P /;
with P .P / the point set of P , such that
(E1) hÃ.P /iDP ;
(E2) for any line L 2 B (seen as a point set), L
Ã
is a line of P .
Of course, from the point of view of Group Theory, no distinction can be made be-
tween a classical quadrangle and its point-line dual – they have the same automorphism
group, cf. the exercise in §1.1.10. But from the viewpoint of Incidence Geometry, there
is indeed a difference: the dual Hermitian quadrangles H.4;q
2
/
D
cannot be fully em-
bedded in a projective space PG.`; q
2
/, where ` 2 N [ f1g. (By “1” we mean any
infinite cardinal number.)
1.4.1 Orthogonal quadrangles. Consider a nonsingular quadric Q of Witt index 2,
that is, of projective index 1,inPG.3; q/,PG.4; q/,PG.5; q/, respectively. So the
only linear subspaces of the projective space in question lying on Q are points and
lines. The points and lines of the quadric form a generalized quadrangle which is
denoted by Q.3; q/, Q.4; q/, Q.5; q/, respectively, and has order .q; 1/, .q; q/, .q; q

2
/,
respectively. As Q.3; q/ is a grid, its structure is trivial.
Recall that Q has the following canonical form:
(1) X
0
X
1
C X
2
X
3
D 0 if d D 3;
(2) X
2
0
C X
1
X
2
C X
3
X
4
D 0 if d D 4;
(3) f.X
0
;X
1
/ C X

2
X
3
C X
4
X
5
D 0 if d D 5, where f is an irreducible binary
quadratic form.