A Course on Elation Quadrangles pot

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 incide

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Claudio Carmeli, Lauren Caston and Rita Fioresi, Mathematical Foundations of Supersymmetry Hans Triebel, Faber Systems and Their Use in Sampling, Discrepancy, Numerical Integration

A Course on

Elation Quadrangles

Koen Thas

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To Caroline

“Les Perspecteurs”

A sketch (85cm  130cm) of the French artist Abraham Bosse (1602–1676) datingfrom 1648, demonstrating the projecting method of Girard Desargues

“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 eralized polygons were classical or dual classical In fact, they obtained this result ingroup theoretical terms (classifying finite split BN-pairs), but Tits remarked the simplegeometrical translation And of course, the converse was already well known In asearch for a synthetic “elementary” proof of the Fong–Seitz result for the specific case

gen-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 anelation point, and a generalized

quadrangle with such a point anelation generalized quadrangle Kantor noticed in the

early 1980s that, starting from a group with a suitable family of subgroups satisfyingcertain properties, one can construct an elation quadrangle from this data in a naturalway, 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 Thisobservation is the precise analogon of the fact that, in a Moufang projective plane, anyline is a translation line, and when one singles out the definition of translation line andtranslation plane, one can also translate the situation in group theoretical terms to agroup with certain subgroups, etc In that case, one obtains a group of order n2with afamily of n C 1 subgroups of order n, two by two trivially intersecting (in the infinitecase, one has to require that the product of any two of these subgroups equals the entiregroup and that the subgroups cover the group) And conversely, starting from suchgroup theoretical data, one readily reconstructs a translation plane for which the groupacts as a translation group The essential difference in this correspondence betweenplanes and quadrangles is that, in the planar case, the translation group necessarily isabelian, and this is not so for elation groups of generalized quadrangles In the planarcase, this property allows one to define a “kernel”, which is some skew field over which

//Translation planes

In this monograph, we will focus on general finite elation quadrangles, so withoutthe 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 elationgroup 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 agiven 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 admitdifferent (t-maximal) elation groups with respect to the same elation point, and theyeven 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 abelianquadrangular case, any i-root and dual i-root involving the translation line or the elationpoint is Moufang, and the unique t-maximal elation group is generated by the Moufangelations In general, such properties do not hold for elation quadrangles We willobtain the first examples of finite elation quadrangles for which not every (dual) i-rootinvolving the elation point is Moufang

So we first have to handle these standard structural questions as a set up for thetheory

After Kantor’s observations, many infinite classes of finite generalized quadrangleswere constructed as elation quadrangles, through the identification of “Kantor families”

in appropriate groups Moreover, up to a combination of point-line duality and Payneintegration, every known finite generalized quadrangleis 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 befound 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 generalized quadranglesare the monographs “Finite Generalized Quadrangles” [44], [46]; “Symmetry in FiniteGeneralized Quadrangles” [68] and “Translation Generalized Quadrangles” [59] (onelation quadrangles with an abelian elation group) These works will only have a smalloverlap with the present notes

I describe, in detail, the beautiful result of Frohardt which solved Kantor’s jecture in the case when the number of points of the (elation) quadrangle is at mostthe number of lines The latter conjecture is the prime power conjecture for elationquadrangles, and states that the parameters of a finite elation generalized quadrangleare powers of the same prime Along the same lines of Frohardt’s proof, I present a niceproof 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 duali-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

con-a more genercon-al setting), con-and his proof is con-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 onmore specific problems, such as a fundamental question posed by Norbert Knarr onthe aforementioned local Moufang conditions (motivated by the idea whether there areother, 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 theliterature Also, about seventy exercises of (usually) an elementary character are for-mulated in the text Exercises which are somewhat less elementary have been indicatedwith a superscript “#”; exercises which come with a superscript “c” ought to be evenmore challenging

Mental note Throughout this work, almost always the generalized quadrangles (and

related objects) we consider arefinite, 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 rangles In particular, I think of the lecture I presented at the conference “FiniteGeometries” 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

quad-x Preface

at the University of Colorado at Denver, USA These talks were often an inspirationfor further research, as were the conversations with members of the audience, such asBill Kantor, Norbert Knarr, Stanley E Payne and Markus Stroppel

A first version of the manuscript was finished during a Research in Pairs stay atthe Mathematisches Forschungsinstitut Oberwolfach, together with Stefaan De Winterand 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 detailedlist 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 theEMS Publishing House for his exceptional good (and pleasant) help in the process ofpublishing this work Finally, during most of the writing, I was a postdoctoral fellow

of the Fund for Scientific Research (FWO) – Flanders

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 C 1/ and their duals 76

8 Elation quadrangles with nonisomorphic elation groups 78 8.1 A nonisomorphism criterion 78

8.2 An example: H.3; q2/, 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  t ) 100

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 gles, for the sake of convenience

quadran-1.1.1 Rank2 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 relation called an “incidence

relation”; so I  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 IL or L Ix

1.1.2 Generalized quadrangles A generalized quadrangle (GQ) of order s; t/ is a

point-line incidence geometry D P ; B; I/ satisfying the following axioms:(i) each point is incident with t C 1 lines (t  1) and two distinct points are incidentwith 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-linepair 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  D P ; B; I/ be a GQ of order s; t/ with s; t 2 N Show that

jP j D s C 1/.st C 1/ and jBj D t C 1/.st C 1/

This exercise shows that if s and t are finite, then jP j and jBj also are In that case,

we call the quadranglefinite If s; t > 1,  is thick; if one of s, t equals 1,  is thin A

thin GQ of order s; 1/ is also called agrid, 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 is also said to be of order s.

There is a parameter-free way to introduce generalized quadrangles, as follows

A rank 2 geometry 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;

(b) every two elements ofP [ 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

Suppose that p; L/ … I Then by projLp we denote the unique point on L collinearwith p Dually, projpL 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 rameters (If D P ; B; I/ is a GQ of order s; t/, D D B; P ; I/ is a GQ of order.t; s/.) Aduality  from the GQ to its dual Dis a map that bijectively sends points

pa-of to lines of D, lines of to points of D, while preserving incidence (This notionwill 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 (defined further in this section) and dualities from  to D

1.1 Elementary combinatorial preliminaries 3

st sC 1/.t C 1/  0 mod s C t:

1.1.5 Collinearity and concurrency Let p and q be (not necessarily distinct) points

of the GQ; 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 2B, we write L M when L and

If two points are not collinear, we also say they are opposite Same for lines A flag is an incident point-line pair.

by S? In particular, let S D fp; qg be a set of two points; then jfp; qg?j D s C 1 or

tC 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??j D s C 1 or jfp; qg??j  t C 1according as p q or p ¦ q, respectively If p q, p ¤ q, or if p ¦ q andjfp; qg??j D t C 1, we say that the pair fp; qg is regular The point p is regular

provided fp; qg is regular for every q 2P n fpg Regularity for lines is defined dually

Exercise Prove that either s D 1 or t  s if has a regular pair of noncollinear points(see [44], 1.3.6)

Anet of order k and degree r is a point-line incidence geometryN D P ; B; I/satisfying the following axioms:

(i) each point is incident with r lines (r  2) and two distinct points are incidentwith 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 Mincident with p and not concurrent with L

We say that k; r/ “are” theparameters of the net Sometimes we also speak of “.k;

r/-net”

Exercise Show that jP j D k2and jBj D kr

Exercise Show that a netN of degree r and order k is an affine plane of order n ifand only if r D k C 1

Theorem 1.1 ([44], 1.3.1) Let p be a regular point of a finite GQ  D P ; B; I/ of

order s; t/, s ¤ 1 ¤ t Then the incidence structure with

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 pwith 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 Theorem1.1

1.1.8 Antiregularity The pair of points fx; yg, x ¦ y, is antiregular if jfx; yg?\

z?j  2 for all z 2 P n fx; yg The point x is antiregular if fx; yg is antiregular for

each y 2P n x?

1.1.9 Triads A triad of points of a GQ is a set of three pairwise noncollinear points.

Let fx; y; zg be a triad of points in a thick finite GQ of order s; s2/ Then jfx; y; zg?j D

sC 1; see [44], 1.2.4 Obviously, jfx; y; zg??j  s 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 of a GQ D P ; B; I/ is

a permutation ofP [ B which preserves P , B and I The set of automorphisms of a

GQ is a group, called the automorphism group of , which is denoted by Aut./ 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 which either is the trivialautomorphism, or it fixes x linewise and has no fixed points inP n x?

about L has the maximum possible order, s, then L is called an axis of symmetry.

Dually, one speaks of acenter 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 Theorem1.6below.)

1.1.12 SubGQs A subquadrangle, or also subGQ,0 D P0;B0; I0/ of a GQ D.P ; B; I/ is a GQ for which P0  P , B0  B, and where I0is the restriction of I to.P0 B0/[ B0 P0/ A subGQ0of order s; t0/ of a finite GQ of order s; t/ iscalled full Dually we define ideal subGQs.

6 1 Generalized quadrangles

The following results will sometimes be used without further reference rems1.4,1.5and1.6can be obtained as easy exercises For the proofs of Theorems1.2and1.3, we refer to [44] In all of the statements below up to Theorem1.6, the gener-alized quadrangles are supposed to be finite

Theo-Theorem 1.2 ([44], 2.2.1) Let0 be a proper subquadrangle of order s0; t0/of the

GQ  of order s; t/ Then either s D s0 or s  s0t0 If s D s0, then each external point of0is collinear with the st0C 1 points of an ovoid of 0; if s D s0t0, then each external point of0is collinear with exactly 1 C s0points of0.

Theorem 1.3 ([44], 2.2.2) Let0 be a proper subquadrangle of the GQ , where 

has order s; t/ and0has order s; t0/(so t > t0) Then the following hold.

(1) t  s; if s D t, then t0D 1.

(2) If s > 1, then t0  s; if t0 D s  2, then t D s2.

(3) If s D 1, then 1  t0< tis the only restriction on t0.

(4) If s > 1 and t0> 1, thenps  t0 s and s3=2 t  s2.

(5) If t D s3=2> 1and t0> 1, then t0Dps.

(6) Let0 have a proper subquadrangle00 of order s; t00/, s > 1 Then t00 D 1,

t0 D s and t D s2.

Theorem 1.4 ([44], 2.3.1) Let0 D P0;B0; I0/be a substructure of the GQ  of

order s; t/ so that the following two conditions are satisfied:

(i) if x; y 2P0are distinct points of0and L is a line of  such that x I L I y, then

L2 B0;

(ii) each element ofB0is incident with s C 1 elements ofP0.

Then there are four possibilities:

(1) 0is a dual grid, so s D 1;

(2) the elements ofB0are lines which are incident with a distinguished point of P ,

andP0consists of those points of P which are incident with these lines;

(3) B0D ; and P0is a set of pairwise noncollinear points of P ;

(4) 0is a subquadrangle of order s; t0/.

The following result is now easy to prove

Theorem 1.5 ([44], 2.4.1) Let  be an automorphism of the GQ  D P ; B; I/ of

order s; t/ The substructure  D P;B; I/of  which consists of the fixed

elements of  must be given by (at least) one of the following:

(i) B D ; and P is a set of pairwise noncollinear points;

(i0) P D ; and B is a set of pairwise nonconcurrent lines;

1.1 Elementary combinatorial preliminaries 7

(ii) P contains a point x so that y x for each y 2 P, and each line ofB is incident with x;

(ii0) Bcontains a line L so that M L for each M 2B, and each point ofPis incident with L;

(iii) is a grid;

(iii0) is a dual grid;

(iv) is a subGQ of  of order s0; t0/, s0; t0 2.

Finally, we recall a result on fixed elements structures of whorls

Theorem 1.6 ([44], 8.1.1) Let  be a nontrivial whorl about p of the GQ D P ; B; I/

of order s; t/, s ¤ 1 ¤ t Then one of the following must hold for the fixed element structure D P;B; I/.

(1) y ¤ y for each y 2 P n p?.

(2) There is a point y, y ¦ p, for which y D y Put V D fp; yg?and U D V? Then V [ fp; yg P  V [ U , and L 2 B if and only if L joins a point of

V with a point of U \P.

(3) is a subGQ of order s0; t /, where 2  s0 s=t  t, and hence t < s.

Exercise Create an “infinite version” of Theorem1.6

1.1.13 Nets and subquadrangles The following theorem is taken from [64] andimplies that a net which arises from a regular point in a thick finite GQ as earlierexplained cannot contain proper subnets of the same degree and different from anaffine plane

Theorem 1.7 ([64]) Suppose that  D P ; B; I/ is a GQ of order s; t/, s; t ¤ 1, with

a regular point p LetNpbe the net which arises from p, and supposeN0

pis an affine plane of order t and s D t2; also, fromN0

pthere arises a proper subquadrangle of  of order t having p as a regular point.

If, conversely,  has a proper subquadrangle containing the point p and of order

.s0; t /with s0 ¤ 1, then it is of order t, and hence s D t2 Also, there arises a proper subnet ofNpwhich is an affine plane of order t.

Proof First suppose that contains a proper subquadrangle 0of order s0; t /, s0; t ¤ 1,containing the point p Then p is also regular in0and since s0 ¤ 1, it follows that

s0 t By Theorem1.3this implies that s0D t and that s D t2 By Theorem1.1, thenetN0

parising from the point p in0is an affine plane of order t , and this net is clearly

a subnet of the net which arises from the point p in

8 1 Generalized quadrangles

Conversely, suppose thatNpis the net which arises from the regular point p in the

GQ, and that it contains a proper subnet N0

pof the same degree In the following,

we identify points of the net with the corresponding spans of points in the GQ, and weuse the same notation

Suppose that P1; P2; : : : ; Pk are the points ofN0

p, define a point setP0 of asconsisting of the points of ŒS

Pi[ ŒS

Pi?, and defineB0

as the set of all lines ofthrough a point ofP0 Then it is not hard to check that the following properties aresatisfied for the geometry0 D P0;B0; I0/, with I0D I \ Œ.P0 B0/[ B0 P0/:(1) any point ofP0is incident with t C 1 lines ofB0;

(2) if two lines ofB0intersect in, then they also intersect in 0

Then by the dual of Theorem 1.4, 0 is a proper subquadrangle of order s0; t /,

s0 ¤ 1, and analogously as in the beginning of the proof, we have that s0 D t and

s D t2

Also, the affine plane of order t which arises from the regular point p in thissubquadrangle is the subnetN0

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  of order s; t/,

s; t ¤ 1, and let Np be the corresponding net If s ¤ t2, thenNpcontains no proper subnet of degree t C 1.

The following corollary tells us that nets which arise from a regular point of a GQand which do not contain affine planes are very “irregular”

Corollary 1.10 ([64]) Let p be a regular point of a GQ  of order s; t/, s; t ¤ 1, and

suppose thatNpis the corresponding net Moreover, suppose that s ¤ t2 If u, v and

ware distinct lines ofNpfor which w ¦ u v, then these lines generate (under the taking of GQ spans) the whole net.

Proof Consider the points of p?n fpg which correspond to the lines u; v; w of Np,and denote them respectively in the same way Then by Theorem1.4, u, v and wgenerate a (not necessarily proper) subGQ0of of order s0; t /, where s0 > 1 ByTheorem1.7this implies that0 D , since s ¤ t2

Hence u, v and w generateNp.



Lemma 1.11 Suppose that  is a GQ of order s; s2/, s ¤ 1, and suppose that0

and 00 are two proper subquadrangles of  of order s Then one of the following

possibilities occurs:

(1) 0\ 00is a set of s2C 1 pairwise noncollinear points (i.e., an ovoid) of 0and

00;

(2) 0\ 00 consists of a point p of0 (and00), together with all lines of0 (and

00)through this point, and all points of0(and00)incident with these lines;

1.2 Some group theory 9

(3) 0\ 00 is a GQ of order s; 1/;

(4) 0 D 00.

Exercise Prove Lemma 1.11 Note that every line of a thick GQ of order s; s2/intersects any subGQ of order s Then use Theorem1.4and a simple counting argument

Theorem 1.12 ([64]) Suppose that  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?n fpg for which q r and q ¦ u.

Then we have one of the following possibilities.

(1) We have that s D t2 and  contains a proper subquadrangle 0 of order t Moreover, if  is not an elation, then0is 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 thatNpis the net which arises from p, and suppose thatN0

pis the (notnecessarily proper) subnet ofNp of degree t C 1 which is generated by u, q and r.Then every point ofN0

p is fixed by  by the previous observation IfN0

p is proper,then by Theorem1.7it is an affine plane of order t and s D t2 Also, there arises aproper subquadrangle0of of order t If  is not an elation, then by Theorem1.6itfollows that there is a proper subquadrangleof order s0; t /, s0 ¤ 1, which is fixedpointwise (and then also linewise) by  Since has a regular point, we have that

s0  t By Theorem1.3,0is necessarily of order t From Lemma1.11now followsthatD 0

IfN0

p D Np, then every point of p?is fixed by  Since  is not the identity, itfollows from Theorem1.6that  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 or1; a group Gwithout 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 theright, such that each element g of G defines a permutation g W X ! X of X and thepermutation defined by gh, g; h 2 G, is given by

ghW X ! X; x7! xg/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

ghD h1gh Thecommutator of g and h is equal to

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 Thecommutator subgroup

of G is ŒG; G, or sometimes G0 Two subgroups A and B centralize each other if

ŒA; B D fidg The subgroup A normalizes B if Ba D B for all a 2 A, which isequivalent with ŒA; B  B

Inductively, we define the n-th central derivative LnC1.G/ D ŒG; GŒn of G asŒG; ŒG; GŒn1, and the n-th normal derivative ŒG; G.n/as ŒŒG; G.n1/; ŒG; G.n1/.For n D 0, the 0-th central and normal derivative are by definition equal to G itself.The series

L1.G/; L2.G/; : : :

is called thelower central series of G If, for some natural number n, ŒG; G.n/D fidg,and ŒG; G.n1/ ¤ fidg, then we say that G is solvable (soluble) of length n If

ŒG; GŒnD fidg and ŒG; GŒn1¤ fidg, then we say that G is nilpotent of class n A

group G is calledperfect if G D ŒG; G D G0

Thecenter of a group is the set of elements that commute with every other element,

i.e., Z.G/ D fz 2 G j Œz; g D id for all g 2 Gg Clearly, if a group G is nilpotent ofclass 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  Z.H /and NM D H Now let M and N be two groups, N Z.N /, M Z.M /, and

W N! M

an isomorphism Then the quotient Q D N M /=K, where K is the normal subgroupf.n; m/ j n 2 N; m 2 M; .n/m D 1g, is the external central product of N and

M provided by the data N; M;  /

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 centralproduct is considered.)

1.2 Some group theory 11

1.2.5 p-Groups and Hall groups For a prime number p, a p-group is a group of

order pn, for some natural number n ¤ 0 ASylow p-subgroup of a finite group G is

a p-subgroup of some order pnsuch that pnC1does not divide jGj Let  be a set ofprimes 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.

AHall -subgroup of a finite group G, where   .G/, and .G/ is the set of

primes dividing jGj, is a subgroup of sizeQ

p2pnp, where pnp denotes the largestpower 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) Gpossesses a Hall -subgroup;

(b) Gacts 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 paqb for some naturalnumbers 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 Pp D ˆ.P /, where Pp D

hwpj 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 aFrobenius group (or G is a Frobenius group in its action on 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 isalmost simple if S  G  Aut.S/, with S a simple

group and Aut.S / its automorphism group

1.3 Finite projective geometry

1.3.1 Projective spaces Below,Fqdenotes the finite field with q elements, q a primepower LetK be any field, and denote by V n; K/ the n-dimensional vector space over

K, n a nonzero natural number If K D Fq 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 asthe 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 theinduced subspace structure IfK D Fqis 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-dimensionalK-vector subspace of V n; K/, it induces a w  1/-dimensionalprojective subspace of V n;K/= D PG.n  1; K/

Exercise LetK D Fq Show that a d (-dimensional)-subspace of PG.n1; q/ contains.qdC1 1/=.q  1/ points In particular, PG.n  1; q/ has qn 1/=.q  1/ points Italso has qn 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”) preserving bijection of the set of subspaces

to itself It can be shown that any automorphism of a PG.n; q/, n 2N and n  2, Fq

a finite field, necessarily has the following form:

W xT ! A.x/T;where A 2 GLnC1.q/,  is a field automorphism ofFq, the homogeneous coordinate

x D x0; x1; : : : ; xn/ represents a point of the space (which is determined up to ascalar), andxD x

0; x1; : : : ; x/ (recall that xiis the image of xi under  ).Here, vectors are identified with row matrices without any further notice Theset of automorphisms of a projective space naturally forms a group, and in case

of PG.n; q/, n  2, this group is denoted by PLnC1.q/ The normal subgroup

of PLnC1.q/ which consists of all automorphisms for which the companion fieldautomorphism  is the identity, is the projective general linear group, and denoted

is the central subgroup of all scalar matrices of GLnC1.q/ Similarly one definesPSLnC1.q/D SLnC1.q/=Z.SLnC1.q//, where Z.SLnC1.q// is the central subgroup

of all scalar matrices of SLnC1.q/ with unit determinant

Anelation 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 Ahomology

1.4 Finite classical examples and their duals 13

either is the identity, or it is an automorphism that fixes a hyperplane pointwise, andone further point not contained in that hyperplane

Exercise Show that each nontrivial elation of PG.n; q/, n 2 N and n  2, has auniquecenter, 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 areknown as the finite “classical generalized quadrangles” (Tits was the first to iden-tify them as generalized quadrangles – see Dembowski [15].) Their point-line dualsare called thedual 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 afull embedding of a rank 2 geometry

D P ; B; I/ in a projective space P, is an injection

W P ,! P P/;

withP P/ the point set of P, such that

(E1) h P /i D P;

(E2) for any line L 2B (seen as a point set), Lis a line ofP

Of course, from the point of view of Group Theory, no distinction can be made tween a classical quadrangle and its point-line dual – they have the same automorphismgroup, cf the exercise in §1.1.10 But from the viewpoint of Incidence Geometry, there

be-is indeed a difference: the dual Hermitian quadrangles H.4; q2/Dcannot be fully bedded in a projective space PG.`; q2/, where ` 2N [ f1g (By “1” we mean anyinfinite cardinal number.)

em-1.4.1 Orthogonal quadrangles Consider a nonsingular quadric Q of Witt index 2,

that is, of projective index 1, in PG.3; q/, PG.4; q/, PG.5; q/, respectively So theonly linear subspaces of the projective space in question lying on Q are points andlines The points and lines of the quadric form a generalized quadrangle which isdenoted by Q.3; q/, Q.4; q/, Q.5; q/, respectively, and has order q; 1/, q; q/, q; q2/,respectively As Q.3; q/ is a grid, its structure is trivial

Recall that Q has the following canonical form:

(1) X0X1C X2X3D 0 if d D 3;

(2) X02C X1X2C X3X4D 0 if d D 4;

(3) f X0; X1/C X2X3 C X4X5 D 0 if d D 5, where f is an irreducible binaryquadratic form

1.4.2 Hermitian quadrangles Next, let H be a nonsingular Hermitian variety in

PG.3; q2/, respectively PG.4; q2/ The points and lines of H form a generalized rangle H.3; q2/, respectively H.4; q2/, which has order q2; q/, respectively q2; q3/.The variety H has the following canonical form (where d D 3 or 4):

The automorphism group of H.3; q2/ is isomorphic to PU4.q/, and PSL4.q/\PU4.q/μ PSU4.q/μ U4.q/ The automorphism group of H.4; q2/ is PU5.q/,and PSL5.q/\ PU5.q/μ PSU5.q/μ U5.q/

1.4.3 Symplectic quadrangles The points of PG.3; q/ together with the totally

iso-tropic lines with respect to a symplectic polarity, form a GQ W q/ of order q

A symplectic polarity ‚ of PG.3; q/ has the following canonical form:

X0Y3C X1Y2 X2Y1 X3Y0:The automorphism group of W q/ is PSp4.q/, while PSL4.q/\ PSp4.q/ μ

Sp4.q/μ S4.q/

1.4.4 Some properties The following results will be very important in this text For

proofs we refer to [44]

Theorem 1.17 ([44], 3.2.1, 3.2.2 and 3.2.3) (i) Q.4; q/ Š W q/D;

(ii) Q.4; q/ Š W q/if and only if q is even;

(iii) Q.5; q/ Š H.3; q2/D.

We sum up the basic combinatorial properties of the classical GQs in the nexttheorem Because of Theorem1.17, we only state the properties for the orthogonalquadrangles and H.4; q2/

Theorem 1.18 (Combinatorial properties, cf [44], §3.3) (i)Let q be odd Then all lines of Q.4; q/ are regular, and all points are antiregular In particular, all spans of noncollinear points have size 2.

(ii)Let q be even Then all lines and points of Q.4; q/ are regular.

(iii)Any line of Q.5; q/ is regular and any span of noncollinear points has size 2 All points are 3-regular.

(iv)For any pair fx; yg of noncollinear points of H.4; q2/, we have jfx; yg??j D

qC 1, while all spans of nonconcurrent lines have size 2.

1.4 Finite classical examples and their duals 15

1.4.5 Small GQs Proofs and references of all the results mentioned in this section

can be found in [44], Chapter 6

Let D P ; B; I/ be a finite thick GQ of order s; t/, s  t

(a) s D 2 By §1.1.4, s C t divides st s C 1/.t C 1/ and t  s2 Hence t 2 f2; 4g

Up to isomorphism there is only one GQ of order 2 and only one GQ of order 2; 4/

It follows that the GQs W 2/ and Q.4; 2/ are self-dual and mutually isomorphic It iseasy to show that the GQ of order 2 is unique

(b) s D 3 Again by §1.1.4we have t 2 f3; 5; 6; 9g Any GQ of order 3; 5/ must beisomorphic to the GQ T2.O/ arising from the unique hyperoval in PG.2; 4/, any GQ

of order 3; 9/ must be isomorphic to Q.5; 3/, and a GQ of order 3 is isomorphic toeither W 3/ or to its dual Q.4; 3/ Finally, there is no GQ of order 3; 6/

(c) s D 4 Using §1.1.4it is easy to check that t 2 f4; 6; 8; 11; 12; 16g Nothing isknown about t D 11 or t D 12 In the other cases unique examples are known, but theuniqueness question is settled only in the case t D 4

Figure 1.4 The generalized quadrangle of order 2; 2/ Its points are the black filled circles; its lines are the straight lines, together with the curves that contain one point lying on the middle of some side of the pentagon, and the two closest points to that point lying in the interior.

The Moufang condition

The Moufang condition is one of the central group theoretical conditions in IncidenceGeometry, and was introduced by Jacques Tits when classifying spherical buildings ofrank at least 3, in his lecture notes “Buildings of Spherical Type and Finite BN-Pairs”[85] It was noted by him that spherical buildings of rank at least 3 satisfy the so-called

“Moufang property”, implying these structures to have a lot of symmetry When therank of these buildings is 2, i.e., when one is dealing withgeneralized n-gons [87], this

is not necessarily the case; many examples exist which are not Moufang (think of thequadrangular or planar case) Already in the 1960s, Tits started a program to obtain allMoufang generalized n-gons, and much later, J Tits and R M Weiss [86] eventuallyfinished the classification of (finite and infinite) Moufang generalized n-gons For thefinite case, this result was already obtained by P Fong and G M Seitz in [16], [17],the most difficult case being the case n D 4 by far, and for this latter case, there isalso a geometrical proof which is a culmination of work by S E Payne and J A Thas[44] (Chapter 9), W M Kantor [28] and the author [61] We refer to the author and

H Van Maldeghem [83] for a survey on old and new results on Moufang generalizedquadrangles We also refer to Chapter 11 of [59], and especially [58] on that matter

In the aforementioned work of S E Payne and J A Thas (and the referencestherein), the importance of local Moufang conditions became obvious – not only nu-merous characterizations of known classes of generalized quadrangles came out; alsothe theory of translation generalized quadrangles essentially arose from it, and theabstraction to elation generalized quadrangles eventually led to many new classes ofgeneralized quadrangles

In this chapter, we review some basic facts concerning the Moufang condition

2.1 Moufang quadrangles

2.1.1 Roots and Moufang roots Note that an ordinary induced quadrangle in a GQ

is just a (necessarily thin) GQ of order 1; 1/ – we call such a subgeometry also an

“apartment” Let A be an apartment of a GQ A root  of A is a set of 5 different

elements e0; : : : ; e4in A such that eiI eiC1(where the indices are taken in f0; 1; 2; 3g),and e0, e4 are theextremal elements of  There are two types of roots, depending

on whether the extremal elements are lines or points; in the second case we speak of

dual roots to make a distinction between the types Also, a (dual) root  without its

extremal elements – theinterior of  – is denoted by P and is called a (dual) i-root.

2.2 Generators and relations 17

Aroot-elation with i-root x; L; y/ D P (and root ) is an element ˛ of Aut./which fixes x and y linewise, and L pointwise We also write ˛ 2 Aut./ŒP, or

˛ 2 Aut./Œ.x;L;y/ So Aut./ŒPis the group of all root-elations with i-root P ByTheorem1.6and the exercise following that theorem, ˛ (if not trivial) cannot fix points

ofP n L, nor lines of B n L? Let U ¤ L and U I u, where u is one of x; y Theroot  and i-root P are calledMoufang if Aut./ŒP acts transitively (and so sharplytransitively) on the points of U n fug

Exercise Show that the definition of Moufang (i-)root is independent of the choice

of U , or u (Do not restrict only to the finite case.) Let be thick and finite, of order.s; t / Show that the root  is Moufang if and only if jAut./ŒPj D s.

2.1.2 Moufang quadrangles A GQ is half Moufang if all its roots, or all its dual

roots, are Moufang It isMoufang if all roots and dual roots are Moufang We already

mentioned that all Moufang quadrangles are classified We will make this more precisefor the finite case later in this chapter For the general case, we refer to [86]

2.2 Generators and relations

Let S ¤ ; be any set (call its elements “letters”), and define thefree group F S/ with alphabet S as follows Put, without loss of generality, S D fsi j i 2 I g, where I

is some nonempty index set, and define another set of letters S1 D fs1

i j i 2 I g.(Note that these are nothing more than symbols!) Aword over the alphabet S is a finite

sequence of letters taken from S [S1; by definition, the “empty word” (a word with noletters) is also a word Define the map red as the map which associates with each word

w over S the word red.w/ which is w in which all occurrences of a sequence of the type

sisi1or s1i si are canceled For instance, if S D fs1; s2g, then red.s1s2s21/D s1 A

reduced word is a word w for which red.w/ D w (it is a fixed element of red) The free group F S/ over S, or with alphabet S, is the set of all reduced words over S,

together with the following operation B; if w, w0are words over S , then ww0denotesthe word over S which is just the concatenation of w and w0, and w B w0 is defined

to be the unique reduced word of fredm.ww0/j m 2 Ng If jSj D n < 1, F S/ iscalled a free group ofrank n.

Exercise Let S, S0 be two sets of the same finite nonzero cardinality Show that

F S /Š F S0/

Exercise Show that a free group of rank 1 is isomorphic toZ; C Show that a freegroup of rank greater than 1 is not abelian

Let G be any group generated by the set S  G; so G D hS i For the sake

of convenience, write S D fsi j i 2 I g Then there is a natural homomorphism

18 2 The Moufang condition

W F S/ ! G defined by

W F S/ ! G; si7! si; i 2 I:

As  is onto, G Š F S /= ker./, where ker./ is the kernel of  So any group

G is the quotient of some free group Motivated by this observation, we can representgroups also as follows Let S be a set of letters, and R a set of words over S Then thegroup which is presented by the generators S and relations R, denoted by hS j Ri, is

the quotient of F S / by the normal subgroup N.R/ generated by the relations of R

So N.R/ is the smallest normal subgroup of F S / containing the words of R (it is theintersection of all normal subgroups of F S / containing the words in R) Suppose that

RD wi/i2IR; then sometimes hS j Ri is also denoted by hS j wiD 1/IRi

Exercise Let n 2 N, n ¤ 0 Then the cyclic group of order n can be presented as

Exercise Show that mij D 2 implies that Œsi; sj D 1 Show that mij D mj i for all

i , j

Recall that adihedral group of rank n, denoted by Dn, is the symmetry group of

a regular n-gon in the real plane (It is also the automorphism group of ageneralized

n-gon [87] of order 1; 1/.)

Exercise Show that the finite Coxeter groups of rank 2 are precisely the dihedral

groups Note that hs1; s2j s2

1 D 1 D s2

2; s1s2/m12 D 1i is isomorphic to Dm12, withm12<1

ACoxeter system is a pair W; S/, where W is a Coxeter group and S the set of

generators defined by the presentation

2.4 BN-pairs of rank 2 and quadrangles 19

2.3.2 Coxeter diagrams Let W; S/ be a Coxeter system Define a (weighted) graph,

called a “Coxeter diagram”, as follows Its vertices are the elements of S If mij D 3,

we draw a single edge between siand sj; if mij D 4, a double edge, and if mij  5, wedraw a single edge with label mij If mij D 2, nothing is drawn If the Coxeter diagram

is connected, we call W; S /irreducible If W is finite, we call W; S/ spherical.

The irreducible spherical Coxeter diagrams (systems) were classified by

H S M Coxeter [13]; the complete list is the following

In this section we consider a quotient group G=N as consisting of cosets of N in G.With this convention, it makes sense to multiply elements of the quotient group with

20 2 The Moufang condition

elements or subgroups of G; it is just the ordinary multiplication of subsets in a group

2.4.1 Tits quadrangles and systems Details on the next theorem can be found in

[59], Chapter 11 For reasons of convenience, we will call a generalized quadrangleadmitting an automorphism group that acts transitively on its ordered ordinary 4-gons,

aTits quadrangle Two flags F D fx; Lg and F0D fy; M g are opposite if x ¦ y and

L¦ M

Proposition 2.1 Let  D P ; B; I/ be a thick Tits generalized quadrangle and let G

be a collineation group of  such that G acts transitively on the set of ordered pairs

of opposite flags of  Let F D fx; Lg be any flag in , with x 2 P and L 2 B, and

let † be an apartment containing F Define B ´ Gx;L, N ´ G†and H D B \ N Then N \ Gxn GL/and N \ GLn Gx/are nonempty Choose arbitrarily sxand sL, respectively, in these sets Then G; B; N / and sx, sLsatisfy the following properties.

The group W is called theWeyl group of G; B; N /, and the elements sx and sL

representatives of the standard generators of W

We now would like to sketch a converse of the previous proposition (All detailscan be found in [59], Chapter 11.) To that aim, we define the notion of a group with a

BN-pair, also called a Tits system.

Let G be a group, and let B and N be two subgroups of G Set H D B \ N Then.G; B; N / is called aTits system of type B2, or B; N / is called aBN-pair of type B2

in G, if there exist elements sxand sLof G, with fsx2; sL2g  H , such that (BN1) up to(BN4) of Proposition2.1hold The group W in (BN2) will be called theWeyl group

and the cosets sxH and sLH thestandard generators of W If is a Tits quadrangle,and G; B; N / is as in Proposition2.1, then we call G; B; N / anatural Tits system

associated with

If G; B; N / is a Tits system of type B2 with Weyl group W and if sx and sLare corresponding representatives of the standard generators of W , then we definethe following incidence structureG;B;N D P ; B; I/ Define Px D hB; Bs xi and

PLD hB; Bs Li, and call these groups maximal parabolic subgroups.

• Points The elements ofP are the right cosets of Px

• Lines The elements ofB are the right cosets of PL

2.4 BN-pairs of rank 2 and quadrangles 21

• Incidence For g; h 2 G, the point Pxg is incident with the line PLh if Pxg\PLh¤ ; (and in this case we may choose g D h)

The group G acts (on the right) as a collineation group on G;B;N, and it actstransitively on the flags, since every flag can be written as fPxg; PLgg (and is theimage under g of the “standard” flag fPx; PLg) If we denote the point Pxby x, thenthe point Pxg can be written as xg Similarly, we write PLas L, and every line can

be written as Lg, for some g 2 G

2.4.2 Bruhat decomposition The Bruhat decomposition states that

w2WBwB:

Consider any flag Fg This flag corresponds to the coset Bg By the foregoingparagraphs we can write g D b0nb, with b; b0 2 B and n 2 N Hence Bg D Bnband so b1fixes F and maps Fginto † It now follows rather easily that the geometry

G;B;Ndoes not contain any digon fx1; x2; L1; L2g, with x1; x2 2 P and L1; L2 2 Bwith xiI Lj, i; j 2 f1; 2g Indeed, by flag transitivity, we may assume x1 D x and

L1 D L By the previous paragraph, we may also assume that the flag fx2; L2g iscontained in †, a contradiction

Suppose thatG;B;N contains a triangle x1I L3I x2I L1I x3I L2I x1, with the xipoints and the Li lines, i D 1; 2; 3 We may again assume x1 D x and L2 D L, and

by the previous paragraph, we may assume that the flag fx2; L1g belongs to † Since

† is a quadrangle, we must have x2 D xs L s x s L But then L3is incident with both xand xsL s x sL

, contradicting the fact that these points are not even collinear

Exercise Show that x and xs L s x s Lare not collinear

Now let x0 be any point not incident with some line L0 We may assume x0 D xand L0 2 † It follows that x0 is incident with at least two lines (the lines through x

in †), that L0is incident with at least two points, and that there is a flag fy; M g with

x0I M I y I L0 The flag fy; M g is unique by the previous paragraph Hence we haveshown thatG;B;N is a GQ

The group G acts as a collineation group, by multiplication at the right, onG;B;N.Since every element of the kernel K of that action must in particular fix B, we see that

K is a normal subgroup of G contained in B Conversely, if K0 E G, and K0  B,then, since the stabilizer of the flag Bg, g 2 G, clearly coincides with Bg, the group

K0stabilizes every flag ofG;B;N, and hence belongs to the kernel K So K is thebiggest normal subgroup of G contained in B, and as such is equal to

g2G

Bg:

The group G=K acts faithfully onG;B;N and the stabilizer of the flag F is B=K If

K D f1g, we say that the Tits system is effective.

22 2 The Moufang condition

Now we determine the stabilizer of † Since N stabilizes † and acts transitively

on the flags of †, it suffices to determine the elementwise stabilizer S of †, and then

NS is the global stabilizer of † So suppose that some b 2 B stabilizes † Since †

is uniquely determined by the opposite flags F and Fsx sLs x sL, this is equivalent with

b2 B fixing Fs x s L s x s L Hence S D B \ Bsx s L s x s L Note that then b automaticallybelongs to Bw, for every w 2 W Hence we can also write

w2W

Bw:

We have proved the following theorem

Theorem 2.2 Let G; B; N / be a Tits system with Weyl group W Then the geometry

G;B;N defined above is a Tits quadrangle Setting

F, and the triple G=K; B=K; NS=K/ is a natural Tits system associated withG;B;N.

The Tits system G; B; N / is calledsaturated precisely when N D NS, with S as

above Replacing N by NS , every Tits system is “equivalent” to a saturated one

2.4.3 BN-Pairs More generally, a group G is said to have a BN-pair B; N /, where

B; N are subgroups of G, if the following properties are satisfied:

The subgroup B, respectively W , is aBorel subgroup, respectively the Weyl group,

of G The natural number n is called therank of the BN-pair (which corresponds to

the rank of the associated “building”) We call the BN-pairspherical if the associated

Coxeter group is finite If the rank is 2 and the BN-pair is spherical, the Weyl groupN=.B\ N / is a dihedral group of size 2m for some positive integer m according tothe list of spherical connected Coxeter diagrams Thetype of the BN-pair is the name

of the corresponding Coxeter group

2.4.4 Classification of finite BN-pairs of rank 2 Using the classification of finite

simple groups, the (finite) groups with a BN-pair of rank 2 can be classified We onlystate the result for type B2BN-pairs

2.4 BN-pairs of rank 2 and quadrangles 23

Theorem 2.3 (Buekenhout and Van Maldeghem [10]) Let G be a finite group with an effective, saturated BN-pair of type B2 Then G is an almost simple group related to one of the following classical Chevalley groups of type B2:

(1) S4.q/Š O5.q/;

(2) U4.q/Š O

6.q/;

(3) U5.q/.

For recent results on finite BN-pairs, we refer the reader to [77], [80], [83], [84]

Exercise# Show that any of the classical quadrangles and their duals has a BN-pair 2.4.5 Split BN-pairs and Moufang quadrangles Let G be a group with a BN-pair

.B; N / of type B2 Put H D B \ N , as before The BN-pair B; N / is calledsplit if

In a celebrated work, P Fong and G M Seitz determined all finite split BN-pairs

of rank 2 (the B2-case being, by far, the most complicated type to handle) We onlystate the result for BN-pairs of type B2

Theorem 2.4 (Fong and Seitz [16], [17]) Let G be a finite group with an effective, saturated split BN-pair of rank 2 of type B2 Then G is an almost simple group related

to one of the following classical Chevalley groups:

(1) O5.q/;

(2) O6.q/;

(3) U5.q/.

Equivalently, a thick finite generalized quadrangle is isomorphic, up to duality, to one

of the classical examples if and only if it verifies the Moufang Condition.

Much more recently the conditions of the previous theorem were relaxed still toobtain the same conclusion The proof is independent of [16], [17]

Theorem 2.5 (K Thas andVan Maldeghem [84]) A thick finite generalized quadrangle

is isomorphic, up to duality, to one of the classical examples if and only if for each point there exists an automorphism group fixing it linewise and acting transitively on the set of its opposite points.

There is an interesting corollary:

Corollary 2.6 (Configurations of centers of transitivity) If a finite generalized

quad-rangle is not (dual) classical, it either has precisely one center of transitivity, a line of centers of transitivity, or no such point at all.

Later, we will meet generalized quadrangles which are not (dual) classical, having

a line of centers of transitivity

Elation quadrangles

We observed in the previous chapter that the classical examples admit interesting tomorphism groups From a local point of view, this motivates us to introduce elationgeneralized quadrangles

au-3.1 Automorphisms of classical quadrangles

We indicate some large automorphism groups of the classical GQs We assume withoutfurther reference the well-known fact that the automorphism groups of the classicalGQs act transitively on the point set (See the last section of this chapter for furthercomments on this property.) By using the explicit forms of the classical GQs, the readercan easily check this fact through straightforward calculation

Observe that .a; b; c; d / stabilizes Q.5; q/ and fixes all lines of Q.5; q/ incident with

x In fact, the abelian automorphism group defined by

ˆD f.a; b; c; d / j a; b; c; d 2 Fqg

3.1 Automorphisms of classical quadrangles 25

acts sharply transitively on the points of Q.5; q/ which are not collinear with x.The hyperplane … with equation X3 D 0 is nontangent, and meets Q.5; q/ in aQ.4; q/-subGQ Clearly, if an element of ˆ maps a point of … \ Q.5; q/ which is notcollinear with x onto another such point, the element fixes …, so also … \ Q.5; q/ Sofor each point x of Q.4; q/, Q.4; q/ admits an automorphism group fixing x linewiseand acting sharply transitively on the points not collinear with x

Symplectic quadrangles Let ‚ be the symplectic polarity of PG.3; q/ associated

with the form

X0Y3C X1Y2 X2Y1 X3Y0:Let W q/ be the GQ defined by ‚ Consider the following element .a; b; c/ ofPGL4.q/ (where a; b; c 2Fq):

.a; b; c/W x0x1x2x3/! x0x1x2x3/

0B

A :

This induces an automorphism of W q/ (the matrix commutes with ‚) and it fixes alllines incident with 0; 0; 0; 1/ in the plane X0 D 0 The set

f.a; b; c/ j a; b; c 2 Fqgforms an automorphism group of W q/ that fixes 0; 0; 0; 1/ linewise (in W q/), andacts sharply transitively on the points of W q/ not collinear with 0; 0; 0; 1/

Hermitian quadrangles Let x ! Nx be the involutory automorphism ofFq2 Definetr.x/ D x C Nx Let U be the unitary polarity of PG.4; q2/ associated to the followingHermitian form

X0Y4C X1Y3C X2Y2C X3Y1C X4Y0;

and let H.4; q2/ be the Hermitian quadrangle corresponding to U Now consider thefollowing element of PGL5.q2/:

0.a; b; c; d /W x0x1x2x3x4/! x0x1x2x3x4/

0BB

A;

with b NbC tr.d C a Nc/ D 0 Then 0.a; b; c; d / preserves U while fixing all linesincident with 0; 0; 0; 0; 1/ in the hyperplane X0D 0 The set

f0.a; b; c; d /j a; b; c; d 2 Fqg

26 3 Elation quadrangles

constitutes an automorphism group of H.4; q2/ that fixes 0; 0; 0; 0; 1/ linewise (inH.4; q2/), and acts sharply transitively on the points of H.4; q2/ not collinear with.0; 0; 0; 0; 1/ (Note that the points of H.4; q2/ which are not collinear with 0; 0; 0; 0; 1/are of the form 1; a; b; c; d /, with b NbC tr.d C a Nc/ D 0.)

Consider the point 0; 0; 1; 0; 0/, which is not a point of H.4; q2/ Then X2 D 0

is the (nontangent) polar hyperplane of 0; 0; 1; 0; 0/ and X2 D 0/ \ H.4; q2/ is aH.3; q2/-quadrangle So for each point x of H.3; q2/, H.3; q2/ admits an automor-phism group fixing x linewise and acting sharply transitively on the points not collinearwith x

The dual Hermitian quadrangles Let q be a prime power For each point of the

GQ H.4; q2/Dthere again is an automorphism group which fixes x linewise and actssharply transitively on the points which are not collinear with x We leave the explicitcalculations to the reader

3.2 Elation generalized quadrangles

We have observed that all finite classical GQs and their point-line duals have, for eachpoint, an automorphism group that fixes it linewise and has a sharply transitive action

on the points which are noncollinear with that point

3.2.1 Elations and quadrangles Let D P ; B; I/ be a GQ If there is an morphism group H of which fixes some point x 2 P linewise and acts sharplytransitively on P n x?, we call x an elation point, and H “the” associated elation group If a GQ has an elation point, it is called an elation generalized quadrangle or,

auto-shortly, “EGQ” We frequently will use the notation x; H / to indicate that x is anelation point with associated elation group H Sometimes we also writex if we donot want to specify the elation group Note that each element of the elation group H

is an elation (and moreover, each element of the subgroup hi also is as such) Notethat we also allow the thin definition of EGQ (although usually we will exclude thistrivial case)

3.2.2 Translation quadrangles If the elation group of an EGQ is abelian, we speak

of atranslation generalized quadrangle (TGQ) The elation group is then usually called

a “translation group”, the elation point a “translation point”

3.3 Maximality and completeness 27

3.3 Maximality and completeness

We will encounter many situations in which a certain group of elations E, say withcenter x, in some quadrangle  D P ; B; I/, is proven to act transitively on the pointsopposite x, in which case the action is sharply transitive So x; E/ is an EGQ Toexpress this fact, we will say that E ist-complete or t-maximal (the “t” stands for

“transitive”) If in a situation it is clear that by “elation group” E of some GQ  withelation point x, we mean “t-maximal elation group” (that is, x; E/ is an EGQ), thensometimes we will omit the terms “t-maximal”/ “t-complete”

3.4 Kantor families

Suppose that x; H / D P ; B; I/ is a finite (not necessarily thick) EGQ of order.s; t /, and let z be a point ofP n x? Let L0; L1; : : : ; Lt be the lines incident with

x, and define ri and Mi by Li I ri I MiI z, 0  i  t Define, for i D 0; 1; : : : ; t ,

Hi D HMi and Hi D Hri, and set J D fHi j 0  i  tg Then we have thefollowing properties:

• jH j D jP n x?j D s2t ;

• J is a set of t C 1 subgroups of H , each of order s;

• for each i D 0; 1; : : : ; t , Hiis a subgroup of H of order st containing Hi as asubgroup

Moreover, the following two conditions are satisfied:

(K1) HiHj \ Hk D f1g for distinct i; j and k;

(K2) Hi\ Hj D f1g for distinct i and j

28 3 Elation quadrangles

Conversely, let H be a group of order s2t (where s; t 2 N and the value 1 isallowed) andJ (respectively J) be a set of t C 1 subgroups Hi(respectively Hi) of

H of order s (respectively of order st ), and suppose that (K1) and (K1) are satisfied

Exercise Show that the H

i are uniquely defined by the Hi

Exercise Show that, for any j 2 f0; 1; : : : ; tg,

j [ S

i¤jHjHin Hj

is a partition of H

We call Hithetangent space at Hi, and J; J/ is said to be aKantor family or

4-gonal family of type s; t/ in H Sometimes we will also say thatJ is a (Kantor,4-gonal) family of type s; t /in H

Notation If J; J/ is a Kantor family in H and A 2J, then Adenotes the tangentspace at A

Let J; J/ be a Kantor family of type s; t / in the group H of order s2t Define

an incidence structure.H; J/ as follows

• Points of.H; J/ are of three kinds:

(i) elements of H ;

(ii) left cosets gHi, g 2 H , i 2 f0; 1; : : : ; t g;

(iii) a symbol 1/

• Lines are of two kinds:

(a) left cosets gHi, g 2 H , i 2 f0; 1; : : : ; t g;

Theorem 3.1 ([26]) If we start with an EGQ x; H /to obtain J as above, then we

have thatx Š .H; J/ So a group of order s2t admitting a 4-gonal family is an elation group for a suitable elation generalized quadrangle.

Exercise Prove thatx Š .H; J/ (Hint: use Figure3.2.) Show that left plication by elements of H guarantees.H; J/ to be an EGQ with elation point 1/and elation group H Let be the isomorphism as defined by the figure Show thatfor y 2P and g 2 H, yg/D g .y//