Francis Buekenhout & Arjeh M. Cohen

Diagram Geometry

related to classical groups and buildings

August 17, 2012

DRAFT

Springer-Verlag

Berlin Heidelberg NewYork

London Paris Tokyo

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Budapest

Contents

1. Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The concept of a geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Incidence systems and geometries . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Subgeometries and truncations . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.5 Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.6 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.7 Permutation groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.8 Groups and geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2. Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.1 The digon diagram of a geometry . . . . . . . . . . . . . . . . . . . . . . . . . 532.2 Some parameters for rank two geometries . . . . . . . . . . . . . . . . . . 582.3 Diagrams for higher rank geometries . . . . . . . . . . . . . . . . . . . . . . 692.4 Coxeter diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762.5 Shadows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822.6 Group diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

2.7 A geometry of type An1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 912.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 982.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

3. Chamber Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1133.1 From a geometry to a chamber system . . . . . . . . . . . . . . . . . . . . 1133.2 Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1203.3 From chamber systems to geometries . . . . . . . . . . . . . . . . . . . . . . 1223.4 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1263.5 The diagram of a chamber system . . . . . . . . . . . . . . . . . . . . . . . . 1303.6 Groups and chamber systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1323.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1393.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

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4. Thin Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1454.1 Being thin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1454.2 Thin geometries of Coxeter type . . . . . . . . . . . . . . . . . . . . . . . . . . 1574.3 Groups generated by affine reflections . . . . . . . . . . . . . . . . . . . . . 1654.4 Linear reflection representations . . . . . . . . . . . . . . . . . . . . . . . . . . 1714.5 Root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1794.6 Finiteness criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1914.7 Finite Coxeter groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1964.8 Regular polytopes revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2014.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2044.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

5. Linear Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2175.1 The affine space of a vector space . . . . . . . . . . . . . . . . . . . . . . . . . 2175.2 The projective space of a vector space . . . . . . . . . . . . . . . . . . . . . 2245.3 Matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2315.4 Matroids from geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2355.5 Steiner systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2415.6 Geometries related to the Golay code . . . . . . . . . . . . . . . . . . . . . 2505.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2585.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

6. Projective and Affine Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2716.1 Perspectivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2716.2 Projective lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2776.3 Classification of projective spaces . . . . . . . . . . . . . . . . . . . . . . . . . 2866.4 Classification of affine spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2916.5 Apartments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2976.6 Grassmannian geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3016.7 Root filtration spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3036.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3176.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

7. Polar Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3277.1 Duality for geometries over a linear diagram . . . . . . . . . . . . . . . 3287.2 Duality and sesquilinear forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 3327.3 Absolutes and reflexive sesquilinear forms . . . . . . . . . . . . . . . . . 3417.4 Polar spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3467.5 The diagram of a polar space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3537.6 From diagram to space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3577.7 Singular subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3617.8 Other shadow spaces of polar geometries . . . . . . . . . . . . . . . . . . 3657.9 Root filtration spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3727.10 Polar spaces with thin lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3827.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385

Contents VII

7.12 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392

8. Projective Embeddings of Polar Spaces . . . . . . . . . . . . . . . . . . . 3978.1 Geometric hyperplanes and ample connectedness . . . . . . . . . . . 3988.2 The Veldkamp space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4028.3 Projective embedding for rank at least four . . . . . . . . . . . . . . . . 4068.4 Projective embedding for rank three . . . . . . . . . . . . . . . . . . . . . . 4158.5 Automorphisms of polar spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 4298.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4358.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438

9. Embedding Polar Spaces in Absolutes . . . . . . . . . . . . . . . . . . . . 4419.1 Embedded spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4429.2 Collars and tangent hyperplanes . . . . . . . . . . . . . . . . . . . . . . . . . . 4469.3 A quasi-polarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4549.4 Technical results on division rings . . . . . . . . . . . . . . . . . . . . . . . . 4599.5 Embedding in 3-dimensional space . . . . . . . . . . . . . . . . . . . . . . . . 4629.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4739.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

10. Classical Polar Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47910.1 Trace valued forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48010.2 Pseudo-quadrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48610.3 Perspective sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49510.4 Apartments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50310.5 Automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50910.6 Finite classical polar spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51410.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52510.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532

11. Buildings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53511.1 Building axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53611.2 Properties of buildings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54511.3 Tits systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55111.4 Shadow spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56011.5 Parapolar spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57211.6 Root shadow spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59211.7 Recognizing shadow spaces of buildings . . . . . . . . . . . . . . . . . . . 59511.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60011.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627

VIII Contents

Preface

In 1955, Chevalley [66] published constructions of simple groups of Lie typefrom Lie algebras. Around that time, Dieudonne [118] had already publishedconstructions of the classical groups and had started to focus on their geo-metric intepretation [119]. The universal geometric counterpart though wasprovided by Tits, whose approach and way of thinking by then began to tran-spire through lectures and papers such as [279]. However, the full extent ofhis constructions as well as the geometric classification of groups of Lie type(at least for rank at least 3) became available through [285]. This is his firstcomprehensive work on buildings, and deals primarily with the classificationof spherical buildings of rank at least three. Later work, using joint work withBruhat, Tits [289] took care of the classification of buildings of affine type andrank at least four (see [305] for an excellent account). Their automorphismgroups are infinite groups, and we will only provide a simple example of thekind of geometry involved. One of the surprising aspects that came forwardfrom Tits work on buildings is the notion of a diagram. Diagrams prescribewhat the geometries underlying the groups of Lie type look like locally. Here,a geometry is an abstract object, little more than a multipartite graph, andthe local information alluded to concerns residues, that is, subgeometries in-duced on the set of vertices adjacent to a set of mutually adjacent verticesof that graph. In the case of bipartite graphs, the geometry is a so-calledgeneralized polygon. As the word indicates, this is a very natural generaliza-tion of polygons. Generalized 3-gons, for instance, are projective planes. Theordinary 3-gon, a combinatorial triangle, is a generalized 3-gon and can beviewed as the smallest projective plane. Moreover, projective planes containhoards of triangles.

In the classification of (non-abelian) finite simple groups, the groups of Lietype play a crucial role, simply because, apart from these and the alternatinggroups, there are only 26 morethe so-called sporadic groups. Buekenhoutlaunched the idea of employing the diagrammatic description of the geome-tries for groups of Lie type to a wider class of groups, preferably one thatwould lead to a classification of the finite simple groups by diagram geometry.By judiciously extending the classes of bipartite graphs allowed as residues,the right diagrams might occur that would fit all simple groups rather thanjust those of Lie type. The idea led to a flurry of activities, ranging from

X Preface

the construction of diagram geometries on the basis of known groups and asystem of their subgroups to the classification of all geometries pertaining toa given diagram. Although quite a few diagrams have been found for finitesimple groups and quite a few interesting classifications of geometries haveseen the light, the classification of finite simple groups has been completedwithout a satisfactory framework offered by diagram geometry. Neverthelessdiagram geometry structured several characterizations of individual sporadicgroups, and provided tools that are useful for geometric alternatives to cer-tain existing parts of the classification. Besides, a lot of finite group theoryis of a very geometric nature, although the proofs are not always formulatedin the associated terminology.

Incidence geometry, though, has beauty in its own right. This is not onlyreflected by quite intriguing diagrams for several simple groups, but also bystriking axioms characterizing spaces related to classical geometries. Mostimpressive is the Buekenhout-Shult description of a polar space by means ofthe single condition that, for each line and each point off that line, either oneor all points of the line are collinear with the point. From this axiom, togetherwith some light nondegeneracy conditions, the full building belonging to anyclassical group distinct from a special linear group and of rank at least two(that is, having a subspace of dimension at least two in the natural repre-sentation space of the group on which the invariant defining form completelyvanishes) can be reconstructed. Besides, whereas diagram geometry functionsbest in cases where ranks are finite, much of the polar space approach remainsvalid for spaces of arbitrary rank. The construction of geometries from spaceswith few axioms is a major theme of diagram geometry. In this respect, theroot filtration spaces form the counterpart of polar spaces. Their axioms aremore involved, but examples exist for all finite groups of Lie type of rank atleast two and distinct from 2F4. In this book, these spaces are introduced andenough properties are derived so as to be able to characterize spaces relatedto projective spaces and to line Grassmannians of polar spaces.

This book provides a self-contained introduction to diagram geometry.The first three chapters are spent on the basic theory. The fourth chaptershows the tight connection with group theory; it deals with thin geometries,which are very close to quotients of Cayley graphs of Coxeter groups. Thesegeometries are abundant in buildings, like the triangles in projective planes.We then treat projective and affine geometry in two chapters. These aregeometries with a linear diagram and linear shadow spaces, which impliesthat they are matroids. This opens the door to variations of the geometriesconnected with buildings. We restrict ourselves to a limited number of vari-ations, just enough to give the flavor of combinatorial structures like Steinersystems in the context of diagram geometry. The last four chapters are de-voted to polar spaces. Their complete classification is found in Tits [285].Here we use a different approach, starting with Veldkamps method [296] ofembedding the polar space in a projective space. There are exceptions, such

Preface XI

as the polar spaces whose projective planes are not Desarguesian, to whichwe devote little attention. This reflects the idea that the book is primarilyan introduction to diagram geometry and the associated synthetic treatmentof fascinating geometric spaces. The intention is to give a flavor of the topicrather than an exhaustive treatment. The references to the literature in theNotes sections are meant to enable the interested reader to further knowl-edge in several directions. The final chapter gives a brief introduction to thetheory of buildings and shows that every spherical building of rank at leastthree is connected with a root filtration space.

The switching of viewpoints within a single geometry by use of the dia-gram leads to axiom systems of various kinds for the classical geometries. Asmentioned above, the root filtration spaces are special among these as everypossible finite group of Lie type acts almost faithfully on such a space. Arepresentative collection of these spaces is directly related to the Lie algebrasintroduced by Chevalley. It is the purpose of the second author to complete asecond volume dedicated to these spaces and the non-classical geometries ofspherical Coxeter type. Preliminary versions of this book, including the in-tended chapters of the second volume, have been available on the internet forover fifteen years. We are very grateful to comments received by enthusiasticreaders and acknowledgments in the form of references to such a volatile siteas the place where the individual chapters were to be found. We hope thereaders will be pleased that at least the first part of the internet versionhas finally been turned into a book.

We thank

(1) the person who has inspired and stimulated us most to write the book:Jacques Tits;

(2) three persons who have greatly contributed: Gabor Ivanyos, Ralf Kohl(ne Gramlich), Antonio Pasini;

(3) the people who have helped in one way or another: Andries Brouwer,Philippe Cara, Hans Cuypers, Chris Fisher, Jonathan Hall, Simon Huggen-berger, Cecile Huybrechts, William Kantor, Jan Willem Knopper, Dim-itri Leemans, Shoumin Liu, Scott Murray, Jos int panhuis, Erik Postma,Ernest Shult, Rudolf Tange, Hendrik Van Maldeghem, Jean-Pierre Tig-nol, David Wales.

(4) Joachim Heinze and Ute Motz at Springer-Verlag for their continuedinterest and infinite patience over the years.

Francis BuekenhoutUniversite Libre de BruxellesBlvd. du TriompheB-1050 BrusselsBelgium

Arjeh M. CohenTechnische Universiteit EindhovenPostbox 5135600 MB EindhovenThe Netherlands

XII Preface

Leitfaden

Elements of diagram geometryChapters 13

Projective spacesChapters 56

Thin geometriesChapter 4

Polar spacesChapters 710

BuildingsChapter 11

Each of the eleven chapters end with a section of exercises and a sectionof notes. Exercises that are deemed hard are indicated by a \ in the margin;those that are deemed computationally intensive by a ].

1. Geometries

The fundamental structure in this book is a geometry. We look at a geom-etry as an incidence system: abstract objects that are related by means ofincidence. The concept of a point lying on a line is carried over to the moreabstract notion of an element of type point being incident to an element oftype line. The basic ideas and definitions are given in this chapter. The usualrelated concepts like homomorphisms and subgeometries, and less generalconcepts such as connectedness and residues, are introduced. The importantnotion of residual connectedness is described in different but equivalent ways(Corollary 1.6.6).

Many of the examples we give display a lot of symmetry. In the latersections of this chapter, we show how the automorphism group of such ahighly symmetric geometry can be used for a complete description of thegeometry in terms of this group and some of its subgroups. Towards the endof the chapter, we describe how properties like residual connectedness can beexpressed in term of these subgroups (Corollary 1.8.13).

1.1 The concept of a geometry

In this book, the word geometry is used in a technical sense, just as words liketopology and algebra. It provides a generalization of the concept of incidence.In a broader context, it would be appropriate to speak of an incidencegeometry, but in this book, there is no danger of confusion.

A geometry consists of elements of different types such as points, lines,planes (or vertices, edges, faces, cells, or subspaces of dimension i where iis an integer). In this context, the reader should momentarily abandon theusual physical viewpoint according to which a line is a set of points, an edgeis a set of two vertices, and so on. The same status will be given to each ofthe different types of elements. Afterwards, we can assign the role of basicelements, traditionally played by points and lines, to any of these types.

Example 1.1.1 See Figure 1.1 for a picture of the cube. Let 1, 2, 3 bethe standard basis of the Euclidean vector space R3 and consider the cube whose 8 vertices are the vectors 1 2 3. The edges (faces) of can be viewed as pairs (respectively, quadruples) of vertices. By replacing

2 1. Geometries

each edge and face by its barycentric vector (up to a suitable scaling by ascalar multiple) in the physical cube, we can visualize by 26 vectors: the 8vectors corresponding to vertices, the 12 vectors 2i 2j (1 i < j 3)corresponding to edges, and the 6 vectors 2i (1 i 3) correspondingto faces. Incidence between elements of can now be visualized as a linesegment connecting two vertices. There are 72 line segments representingincident pairs of elements and 48 triangles representing incident triples.

Fig. 1.1. The cube geometry

In general, incidence is a symmetric, reflexive relation on the set of ele-ments with no two elements of the same type incident. The rank of a geometryis the number of distinct types of elements. This number is almost alwaysassumed to be finite. We use this word rather than dimension because thelatter has several classical meanings which might not agree with this defini-tion. We often use natural numbers for types; otherwise, types are written ina special font.

Example 1.1.2 The Euclidean affine plane E2 provides a rank two geometrywith types point and line which we call the real affine plane. The Euclideanaffine space E3 provides a rank three geometry with types point, line, andplane which we call the real affine geometry of rank three. The Euclideandistance plays no role here.

Example 1.1.3 A polygon in the Euclidean plane is loosely defined as a2-dimensional simply connected shape whose boundary is made up of a finitenumber of points, called vertices, and line segments, called edges, such thateach vertex is on exactly two line segments. Similarly, a polyhedron in theEuclidean space is a 3-dimensional simply connected shape whose boundaryis made up of a finite number of vertices, edges (line segments again), andpolygons in planes (called faces), such that each edge is on exactly two faces.It provides a rank three geometry whose types are vertex, edge, face. Well-known examples are pyramids, prisms, the tetrahedron, and the octahedron.

1.1 The concept of a geometry 3

Figure 1.2 illustrates two other famous examples: the icosahedron and thedodecahedron. These are dual to each other in the sense that, replacing thenames vertex, face by face, vertex, respectively, we obtain each geometryfrom the other.

Fig. 1.2. Icosahedron and dodecahedron

Example 1.1.4 An edge-to-edge tiling or tessellation of E2 by polygonsgives a rank three geometry whose types are vertex, edge, face. The familiarpattern of a brick wall is not edge-to-edge. Well-known examples that we willcome across later are the three regular tessellations with equilateral trian-gles, squares, and regular hexagons, respectively. Figure 1.3 gives one moretessellation.

Fig. 1.3. A tiling of the Euclidean plane

4 1. Geometries

Example 1.1.5 A face-to-face tessellation of E3 by polyhedra provides arank four geometry whose types are vertex, edge, face, cell. The typicalexample is built up from cubes (as cells). Figure 1.4 depicts a less obviousexample, in which all cells are copies of a so-called truncated octahedron.

Fig. 1.4. A tiling of Euclidean space by truncated octahedra

Remark 1.1.6 There may be more than one way to view a familiar geomet-ric object as a geometry. The cube, for instance, is a rank three geometry inthe guise of a polyhedron, but a rank two geometry when the faces are disre-garded. Also the polyhedron with 26 vertices (the vectors), 72 edges (the linesjoining vectors representing incident elements of the cube), and 48 trianglesof Figure 1.1 can be interpreted as the rank three cube geometry.

1.2 Incidence systems and geometries

In this section we give the formal definition of a geometry. It is close to thenotion of a multipartite graph.

For the duration of the section, I will be a set, called the type set. Itneed not be finite. Its elements as well as its subsets are called types.

Definition 1.2.1 A graph is a set of vertices (singular: vertex) with afamily of (unordered) pairs of distinct vertices, called edges. Two vertices x,y are called adjacent if {x, y} is an edge; we also express this fact by sayingthat y is a neighbor of x.

A graph is fully determined by the pair (V,E), where V is the vertex setand E is the edge set, but also by (V,), where V is as before and is theadjacency relation. For we use infix notation, so x y means the same as{x, y} E.

1.2 Incidence systems and geometries 5

A graph is called bipartite if its vertex set can be partitioned into twonon-empty subsets, X1 and X2, such that there are no edges inside X1 norinside X2. More generally, a graph with vertex set X is called multipartitewith partition (Xi)iI if X is partitioned into non-empty subsets Xi (i I)such that, for each i I , there are no edges inside Xi.

A complete graph is a graph in which all pairs of vertices are edges. Apartial subgraph of a graph (V,E) is a graph (V , E) with V V andE E. If A is a subset of V , then the subgraph of (V,E) induced on Ais the pair (A,E (AA)). If the subgraph is a complete graph, then A isoften referred to as a clique.

Definition 1.2.2 A triple = (X, , ) is called an incidence system overI if

(1) X is a set (its elements are also called elements of );(2) is a symmetric and reflexive relation on X ; it is called the incidence

relation of , usually written as an infix operator; so, for x, y X ,incidence between them is denoted x y;

(3) is a map from X to I , called the type map of , such that distinctelements x, y X with x y satisfy (x) 6= (y); members of the pre-image 1(i) are called elements of type i, or i-elements.

In an incidence system = (X, , ) over I , the set X is the disjointunion of the sets Xi =

1(i), the inverse image of i under , for i I .Thus, (X, ) is a multipartite graph with partitioning (Xi)i(X). (The typesin the codomain I of that are not in the image (X) play an insignificantrole; we usually have I = (X).) The concept is illustrated by Figure 1.5,where a picture of a geometry associated with the tetrahedron is drawn. Here,incidence is represented by edges; incidence of an element with itself is notdrawn.

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6 1. Geometries

For an incidence system over I , in which is surjective, we also write((Xi)iI , ) instead of (X, , ), suppressing . Observe that X =

iI Xi

and that (Xi) = {i} for each i I , so that indeed is uniquely determinedby the pair ((Xi)iI , ). The incidence relation on each Xi is the identity,so the new description of is close to a multipartite graph but differs fromit in two ways:

(1) Although the parts Xi are distinguished, the type map does not existon a multipartite graph and is made explicit in .

(2) The vertices of the multipartite graph are not adjacent to themselves butare incident in .

Definition 1.2.3 Let = (X, , ) be an incidence system over I . The setI is called the type of and the cardinality of I is called the rank of ,and denoted by rk ( ).

If A X , we say that A is of type (A) and of rank |(A)|, the car-dinality of (A), denoted by rk (A). The corank of A is the cardinality ofI\(A).

Given Y X , we write Y for the set of all elements incident with everymember of Y . For x X , we sometimes write x Y instead of x Y . Wealso write x instead of {x}.

The graph with vertex set X whose edges are the unordered pairs {x, y}of distinct vertices x, y with x y is called the incidence graph of .

A flag of is a set of mutually incident elements of . Flags of of typeI are called chambers.

The type of the set X of elements of the incidence system coincideswith I , the type , if and only if the type map is surjective.

In an incidence system over I , the restriction of the type map to everyflag of is an injection. For, a flag of has at most one element of eachtype.

If denotes the relation on X of being distinct and incident, then, withthe notation of Definition 1.2.1, the incidence graph is (X,). Moreover, flagsare cliques.

The 48 chambers of the cube of Example 1.1.1 are displayed as the trian-gles in Figure 1.1.

The word flag is inspired by the visualization in Figure 1.6 of a flag ina projective geometry but can be based on the real affine space of Example1.1.2 as well. The point of the flag is the ball at the top, the line of the flagis the pole with that top, and the plane is the cloth of the flag. A typicalexample of a flag in E3 is a triple consisting of a point, line, and plane suchthat the point is on the line and the line is on the plane (because then thepoint is also incident with the plane).

1.2 Incidence systems and geometries 7

Fig. 1.6. A flag, consisting of an incident point, line, and plane

Remark 1.2.4 By Zorns Lemma, every flag is contained in at least onemaximal flag, that is, a flag not properly contained in any other flag.

In an incidence system, chambers are maximal flags. In general, however,the converse does not hold: Exercise 1.9.3 gives examples of incidence systemsover I in which maximal flags exist that do not have type I .

Definition 1.2.5 Let be an incidence system over I . If every maximal flagof is a chamber, then is called a geometry over I .

In terms of the incidence graph, an incidence system is a geometry if andonly if the image under the type map of every maximal clique is I . In thiscase, the type map is surjective.

Example 1.2.6 Figure 1.7 shows the usual picture of a tetrahedron in Eu-clidean affine space E3 and Figure 1.5 represents the geometry by a polyhe-dron in E3 having 4 + 6 + 4 = 14 vertices (elements) and 4 3 2 = 24triangular faces (chambers of the geometry).

As an abstract geometry, it is the special case n = 4 of the geometry(X1, . . . , Xn1, ) over [n 1] that we define below for each n > 1. Here andelsewhere, [n] denotes the set {1, . . . , n}, and

(Xk

), for a set X and a number

k, the collection of subsets of X of size k. Now Xi =([n]i

)for i [n 1] and

is defined by U W if and only if either U W or W U . It has 2n 2elements and n! chambers.

The condition defining the incidence relation is called symmetrized in-clusion. More generally, the symmetrized version of a relation will be therelation given by x y if and only x y or y x.

Definition 1.2.7 A geometry is firm (respectively, thick) if every flag oftype other than I is contained in at least two (respectively, three) distinctchambers of . It is called thin if every flag of type I\{i} for some i I iscontained in exactly two chambers of .

8 1. Geometries

Fig. 1.7. The tetrahedron

Example 1.2.8 A polygon in E2 is a familiar rank two geometry. It is thin.Its incidence graph is again a polygon, with twice as many vertices as theoriginal. Let n N, n 2. The polyhedron in the Euclidean affine space E3for n = 3, and, more generally, the polytope in En, are thin geometries ofrank n.

Remark 1.2.9 Although we almost always work with firm geometries, wegive two cases where non-firm geometries naturally occur. First, the unfoldingof a polyhedron in an elementary and practical view, as depicted in Figure1.8. This is a geometry with 14 vertices, 19 edges, and 6 faces, which is notfirm. Second, a chamber, viewed as a subgeometry of a geometry (Definition1.4.1), is clearly not firm.

Fig. 1.8. An unfolding of the cube

1.3 Homomorphisms

In geometry, as in every structure theory, the concepts of homomorphism,isomorphism, and automorphism are essential. We also need homomorphismsof a more general kind, called weak homomorphisms. Fix a type set I and anincidence system = (X, , ) over it.

1.3 Homomorphisms 9

Definition 1.3.1 Let = (X , , ) be an incidence system over I . Aweak homomorphism : is a map : X X such that, for allx, y X ,(1) x y implies (x) (y);(2) (x) = (y) if and only if ((x)) = ((y)).

If, in addition, I = I and (x) = ((x)) for all x X , then is called ahomomorphism.

Properties like injectivity applied to the weak homomorphism will beunderstood to apply to the underlying map X X . An injective homomor-phism : of incidence systems is also called an embedding of into .

A bijective weak homomorphism whose inverse 1 is also a weak homo-morphism is called a correlation. If is a homomorphism and a correlation,then we call an isomorphism and write = . In this case 1 is alsoan isomorphism.

The correlations of onto itself, called auto-correlations, form a group,denoted by Cor( ), whose multiplication is composition of maps. The isomor-phisms of onto itself, called automorphisms, also form a group, denotedAut( ).

Remark 1.3.2 A weak homomorphism : preserves incidence andsends elements of the same type in I to elements of the same type in I .If is surjective, then induces a map I : I I . In case = andI = I , the restriction of the map 7 I to Cor( ) is a homomorphismCor( ) Sym(I) of groups, with kernel Aut( ). Here, Sym(X) denotesthe group of all permutations of a set X , where a permutation of X is abijection X X . For, an auto-correlation of is an isomorphism if andonly if it fixes types, that is, is the identity on I . In particular, the groupAut( ) is a normal subgroup of Cor( ).

Example 1.3.3 Let be the rank two geometry consisting of the 20 verticesand 30 edges of the dodecahedron D, drawn in the right hand side of Figure1.2. Identifying opposite elements of D we obtain a natural homomorphism from onto the famous Petersen graph Pet with 10 vertices and 15 edges,depicted twice in Figure 1.9.

This can be verified by labelling the vertices of D with ordered pairs (i, j) fordistinct i, j [5], in such a way that the vertices labelled (i, j) and (k, l) forman edge if and only if there is an even permutation of [5] moving (i, j, k, l)to (1, 2, 3, 4). This labeling is given in Figure 1.10. The Petersen graph canbe described as the vertex set {{i, j} | 1 i < j 5} with {i, j} and {k, l}joined by an edge if and only if {i, j} {k, l} = . In this set-up, there is ahomomorphism : Pet determined by ((i, j)) = {i, j}.

10 1. Geometries

Fig. 1.9. The Petersen graph in two guises

The map interchanging (i, j) and (j, i) for all i, j clearly induces an auto-morphism z of order two of. The group Aut() is a direct product C2Alt5of the cyclic group C2 of order two generated by z (in general, Cn denotesthe cyclic group (that is, a group generated by a single element) of order n)and the alternating group Alt5 (that is, the group of all even permutations of[5]). The group Aut(Pet) is isomorphic to the symmetric group Sym5 on fiveletters. The element z maps onto the identity on Pet, but new automorphisms(induced by odd permutations) appear.

21

54

32

14

35

42

13

15

41

23

24

25

12

53

34

31

5145

43

52

Fig. 1.10. Labelling the vertices of a dodecahedron

Example 1.3.4 The homomorphism : Pet of Example 1.3.3 mapsthe 12 faces of D into six circuits of length 5. There are 12 circuits of length 5on the Petersen graph falling into two classes: those coming from a face of Dand the six others (each coming from a shortest path in D from a vertex toits opposite). Two members of the same class share a unique edge. Automor-phisms coming from odd permutations in Sym5 interchange the two classes,while those coming from even permutations (Alt5) preserve them. The rankthree geometry consisting of the vertices and edges of Pet and one of theseclasses of circuits of length 5 is called a hemi-dodecahedron. The map provides a homomorphism from D onto the hemi-dodecahedron built fromone of the two classes. But symmetry suggests that, for the other class ofsix circuits of length 5, there must be another rank three geometry isomor-phic to D and a surjective weak homomorphism from it onto the other

1.3 Homomorphisms 11

hemi-dodecahedron on Pet. This geometry is realized by the great stellateddodecahedron whose vertices are the same as those of the dodecahedron butwhose faces are star-shaped pentagons (so-called pentagrams). See Figure1.11.

Fig. 1.11. The great stellated dodecahedron

Definition 1.3.5 Let A be a permutation group on a set X , that is,a subgroup of the group Sym(X) of all permutations of X . A minimal A-invariant subset of X is called an A-orbit of X . The collection of all A-orbitsis denoted by X/A.

Suppose that A is a group of automorphisms of . The quotient inci-dence system of by A is the incidence system /A = (X/A, /A, /A)over I , where two orbits Ax and Ay of x, y X , respectively, are incidentwith respect to /A if there are x Ax and y Ay with x y, and /Amaps Ax to (x).

More generally, let E be an equivalence relation defined on X such that(x, y) E implies (x) = (y) whenever x, y E. The quotient incidencesystem of by E is the incidence system /E = (X/E, /E, /E) over I ,where X/E is the set of equivalence classes of E, two classes e, f are incidentwith respect to /E if there are x e and y f with x y, and /E maps eto (x) for x e.

Indeed, /E is an incidence system over I and /A is the special case of/E, where E is defined by (x, y) E if and only if the A-orbits of x and ycoincide.

The map sending an element to its equivalence class under E defines asurjective homomorphism from to /E, and so this quotient is also theimage of an incidence system homomorphism in the sense of Definition 1.3.1.

12 1. Geometries

Example 1.3.6 The hemi-dodecahedron of Example 1.3.4 is the quotient ofthe dodecahedron (viewed as a rank three geometry) by the group of ordertwo generated by z.

We will construct more examples of this kind, starting from the followingthin geometry := (Y1, Y2, . . . , Yn, ), where n N. For j [n], an elementof Yj is a vector x of R

n of the form x =

iJ ixi, where J ([n]j

)and xi

{1} for all i J . For subsets J ([n]j

)and K

([n]k

)with 1 j k n,

the elements x iJ i and y

iK i, are incident if J K andy x iK\J i; now, is the symmetrization of this relation. The setYj has size 2

j(nj

)and there are 2nn! chambers. The special case n = 3 is the

geometry of the cube of Example 1.1.1.The group A generated by the scalar multiplication by 1 is a group

of automorphisms of of order two. The quotient incidence system /A isagain a geometry.

The group B of all sign changes in coordinates, is also a group of auto-morphisms of of order 2n. The quotient incidence system /B has a singleelement of type n, which is incident with every element of the geometry.

Example 1.3.7 Start with a triangle as a rank two geometry. See Figure1.12.

B C B=Ca

b

a

A A

c

B C

b

A

b=cc

a

Fig. 1.12. The folding of a triangle

At the left hand side, the usual picture has been drawn and at the righthand side a picture that represents the incidence graph after removal of thedirected edges. The points labelled A, B, C represent the vertices and thepoints labelled a, b, c represent the edges of the triangle. A folding of thetriangle in the physical sense is depicted in the incidence graph by means ofdirected edges. The equivalence relation in which two elements are equivalentwhenever the folding moves one to the other can be used to describe thephysical folding as a surjective homomorphism. The folded geometry appearsin the middle of Figure 1.12.

1.3 Homomorphisms 13

Definition 1.3.8 In Remark 1.3.2 we saw that every auto-correlation of induces a permutation I of I . If I has order two, then is called aduality. If both I and have order twp, then is called a polarity. Ifboth I and have order 3, then is called a triality.

Example 1.3.9 Polarities and dualities in projective planes are classical ex-amples of auto-correlations. For those (as yet) unfamiliar with projectiveplanes, we describe the simplest case. Consider once more the triangle repre-sented by the hexagon at the right hand side of Figure 1.12. A rotation over60 degrees around the center results in a correlation. Its square is a rotationover 120 degrees, which is a nontrivial automorphism. Therefore, the rotationover 60 degrees induces a duality that is not a polarity. Rotation over 180degrees leads a polarity.

Fig. 1.13. A regular tessellation with labelled vertices

Example 1.3.10 Consider the regular tessellation of E2 by triangles indi-cated in Figure 1.13. The vertices can be labelled by 1, 2, 3 in such a waythat the labels of the three vertices of each triangle are distinct. Let bethe geometry over [3] whose elements of type i are the vertices labelled i foreach i [3] and in which two elements are incident if they are vertices ofa common triangle. It is easy to spot a triality sending elements of types 1,2, 3 onto elements of types 2, 3, 1, respectively. There are also polarities, soCor( )/Aut( ) is isomorphic to Sym3.

Example 1.3.11 The unit geometry over I is the geometry (I, , ) overI where is the complete relation (every two elements are incident) and is the identity map on I . For every geometry over I the assignment oftypes is a homomorphism from onto the unit geometry over I . Similarly,every permutation of I determines a correlation of the unit geometry over Iand every injective map from I into a set I is a weak homomorphism of theunit geometry over I into the unit geometry over I .

14 1. Geometries

1.4 Subgeometries and truncations

A polyhedron can be seen as a subgeometry of E3 consisting of a selectionof the points, lines, and planes, if no degeneracies like the one depicted inFigure 1.14 occur.

b

c

a f d

e

Fig. 1.14. A hexagon in the plane

In this section we formalize the notion of subgeometry as well as trunca-tion, which is the geometry that remains after the removal of all elements ofcertain designated types. We also introduce some of the geometries of our coreinterest, like projective and affine geometries related to a finite-dimensionalvector space. Fix two sets of types, I and I , let = (X, , ) be an incidencesystem over I , and let = (X , , ) be an incidence system over I .

Definition 1.4.1 If X X , (where both incidence relations areviewed as subsets of X X), I I , and : X I is obtained byrestricting to X , then is a partial subsystem of over I . In most (ifnot all) applications of this concept, we will deal with the particular case ofa subsystem, where is the restriction of to X X . In that case isdetermined by the choice of X in X up to the choice of I within I . If thechoice is minimal, that is I = (X ), then (X , , I ) is called the subsystemof induced on X .

A subsystem of that is a geometry is called a subgeometry of .

Remark 1.4.2 The hexagon in the Euclidean plane shown in Figure 1.14 isa clear example of a partial subsystem that is not a subsystem: the incidenceof point d with the Euclidean line af does not occur in the hexagon.

The subsystem of induced on a flag of type F is isomorphic to the unitgeometry over (F ).

Exercise 1.9.6 shows that the property of being a subsystem can also bedescribed in terms of an injective homomorphism.

Definition 1.4.3 Let A be a group of automorphisms of . The fixed sub-system of with respect to A is the subsystem induced on the set of elementsof that are fixed by A.

1.4 Subgeometries and truncations 15

Example 1.4.4 Consider the rank two geometry of the triangle with points{a, b, c} and edges {A,B,C} such that a, b C, b, c A, and a, c Btake care of all incidences. The map inducing (b, c)(B,C) on the element sethas fixed elements {a,A}, so the fixed subsystem with respect to the group{id, } has rank two. But neither {a} nor {A} extends to a flag of rank two,so the fixed subsystem is not a subgeometry of the triangle.

Example 1.4.5 The complex affine plane can be viewed as a rank two geom-etry in much the same way as the real affine plane of Example 1.1.2. Take Ato be the group of order two consisting of complex conjugation and the iden-tity, acting coordinate-wise on the complex affine plane. The fixed subsystemwith respect to A is a subgeometry; it is the real affine plane E2 discussed inExample 1.1.2.

In Section 1.1, we mentioned that a cube can be seen both as a polyhedronand as a graph. Similarly, the Euclidean space E3 can be seen as a rank threegeometry but also as a rank two geometry (for instance by disregarding itsplanes). These are instances of the general concept of truncation.

Definition 1.4.6 Let J be any subset of I . The J-truncation of is thesubsystem of over J on the set 1(J). It is denoted J .

Thus, an incidence system of rank n allows for 2n truncations. If is ageometry over I , and J I , then J is a geometry over J , so J coincideswith the subgeometry induced on 1(J).

If, in Figure 1.5, we delete the elements of type face, the picture of the{vertex, edge}-truncation of the tetrahedron arises.

Example 1.4.7 We construct examples of division rings, also known asskewfields. These are (associative but not necessarily commutative) ringswith a unit in which every non-identity element has a multiplicative inverse.These division rings will be used as scalars for vector spaces, from whichprojective and affine geometries will emerge. Well-known non-commutativedivision rings are the quaternions; see Exercise 1.9.11(b). Here we introduceanother series of division rings.

Let K be a field admitting an automorphism of order three. The fixedpoints of in K form a subfield F of K and K can be viewed as a vector spaceover F. The dimension of K over F is equal to 3 (proving this fact is part ofExercise 1.9.12). Fix a F. The associative algebra D over K is defined asthe left vector space D = K Kj Kj2 with the multiplication rules

jx = (x)j for x K, and j3 = a.

16 1. Geometries

As is not the identity, the multiplication is not commutative, so D cannotbe a field. We show that, for suitable choices of a, the ring D is a division ring.We proceed in three steps, the first two of which are a general construction.

Step 1. The subfield F is the center of D . For each k K\F, its centralizerin D coincides with K, that is, {x D | xk = kx} = K.For, if x = x0 + x1j + x2j

2 D with x0, x1, x2 K, then xk = kx implies

(k (k))x1 = (k 2(k))x2 = 0.

As (k) = k or 2(k) = k conflicts with k 6 F, the left factors of x1 and x2are nonzero, hence invertible, elements of K, so we find x1 = x2 = 0. Thisproves x = x0 K, so the centralizer in D of k is contained in K. As theother inclusion is obvious, Step 1 is complete.

We need the norm map N : K F of K over F, which is given byN(x) = x(x)2(x) for x K. An important property of this map is thefact that it is multiplicative: N(xy) = N(x)N(y) for all x, y K. We areinterested in the case where N is not surjective and, in fact, a is not a norm.

Step 2. If a F\N(K), then D is a division ring.Assume that D is not a division algebra. This means there is a nonzerox D such that xD 6= D . Let

V = {y D | xy = 0}.

This is the kernel of the linear map D D given by y 7 xy, where Dis viewed as a right vector space over K. By standard linear algebra, wefind dim K(V ) + dim K(xD ) = dim K( D ) = 3, so either dim K(V ) = 1 anddim K(xD ) = 2, or dim K(V ) = 2 and dim K(xD ) = 1. For each nonzero y V we have yD V , hence dim K(yD ) = 1 if dim K(xD ) = 2. Substitutingy for x if necessary, we may therefore assume dim K(xD ) = 1, so xD = xK.This implies that there is some k K with xj = xk. The ensuing equationx(j k) = 0 shows that right multiplication by j k is a singular linear mapD D of left vector spaces over K. Its matrix with respect to the basis 1,j, j2 is

k 1 00 (k) 1a 0 2(k)

.

The matrix is singular and its determinant is N(k) + a, so a = N(k) is thenorm of k.

Step 3. The setting of Step 2 is realized by the following choices.

(1) F = Q.(2) K = Q(u), where u3 + u2 2u 1 = 0 (this equation is satisfied by the

algebraic number 2 cos(2/7)).

1.4 Subgeometries and truncations 17

(3) a = 2.(4) Aut(K) determined by (u) = u2 2.

We verify the hard part of the proof that this is an example, namely thata 6 N(K). Suppose that x = x0 + x1u + x2u2 K, with x0, x1, x2 Q,satisfies N(x) = 2. Let g be the minimal positive integer with gxi Z foreach i {0, 1, 2}. A straightforward computation shows

N(x) = x30 x20x1 + 5x20x2 2x0x21 + 6x0x22x0x1x2 + x31 x21x2 2x1x22 + x32.

Applying this to gx, taking values modulo two, and expanding the right handside of the equation below, we verify that

0 = g3N(x) 1 + (gx0 + 1)(gx1 + 1)(gx2 + 1) (mod 2).

If some gxi would be odd, then the product of three terms at the righthand side vanishes and the equation 0 1 (mod 2) results, a contradiction.Therefore, each gxi is even, so gxi = 2mi for certain integers mi (i = 0, 1, 2).Moreover, by minimality of g, the number g is odd (if it were even, then thehighest power of 2 dividing g would be equal to the highest power occurringin the denominator of some xi and so gxi would be odd, a contradiction).Now, writing m = m0 + m1u + m2u

2, we have 2g3 = g3N(x) = N(gx) =N(2m) = 8N(m) 8Z, which contradicts that g is odd. Hence, there is nox K with N(x) = 2.

Remark 1.4.8 Let D be a division ring. Since D need not be commutative,it is necessary to specify, for a vector space over D , whether it is a right or aleft vector space, that is, whether we take the scalar multiplication as a rightor a left action of D on V . We will mainly restrict ourselves to right vectorspaces over D . The reason is that most group actions in this book, includingthe action of the group GL(V ) of linear transformations of V , are on the left,so that linearity can be nicely expressed as the associativity rule

g(v) = (gv) (g GL(V ), v V, D ).

The mathematical consequences of this decision are minor, as the right scalarmultiplication of D is equivalent to the left scalar multiplication of the op-posite division ring, as explained in Exercise 1.9.11.

Example 1.4.9 Let n N, n 1, and let V be a vector space of finitedimension n + 1 over a division ring. The projective geometry PG(V ) isdefined as follows.

(1) The elements are all nonempty subspaces of V except {0} and V itself.(2) Subspaces U and W are incident if and only if either U W or W U .

18 1. Geometries

(3) The type of an element is its affine dimension (that is, its dimensionas a vector space).

This incidence system is a geometry over [n]. Its elements of type 1 (respec-tively, 2) are usually called points (respectively, lines). The [2]-truncationof PG(V ) is closely connected to the projective space on V , which will beintroduced in Definition 5.2.1. These spaces will be studied extensively inChapters 5 and 6.

Here is a similar construction of affine geometries.

Example 1.4.10 Let n N, n 1, and let V be a vector space of finitedimension n over a division ring. The affine geometry AG(V ) is defined asfollows.

(1) The elements are all nonempty affine subspaces of V except V itself.(2) Incidence is defined by symmetrized inclusion (see Example 1.4.9(3)).(3) The type of an element of affine dimension i is equal to i+ 1.

This incidence system is a geometry over [n]. Its elements of type 1 (respec-tively, 2) are usually called points (respectively, lines). The [2]-truncation ofAG(V ) is closely connected to the affine space on V , which will be introducedin Proposition 5.1.3. These spaces will be studied extensively in Chapters 5and 6.

Remark 1.4.11 Consider the case where n = . We recall that if c is acardinality with c n, then c+n = n and also that the class of all cardinalitiesis well ordered. Suppose we define AG(V ) as above, with incidence beingsymmetrized inclusion. Letting the type be dimension would be a bad choice:all affine subspaces of finite codimension would have the same type. It seemsmore appropriate to define the type (S) as the pair (d, c) where d = dimS,c = codim S. Observe that d+ c = n, hence at least one of d, c must be equalto n. Unfortunately, AG(V ) as defined in this very natural way, is not even anincidence system because there exist elements with the same type d = n = cwhich are contained in each other and which are therefore incident.

There is a natural total order on the set I of types: put (d, c) (d, c)if and only if d d and c c. For, given two types (d, c), (d, c), thend+ c = n = d + c and if d < d, then d < n, whence c = n and so c c. Ifwe want to force a geometry AG(V ) despite the preceding observations, wecan truncate and keep as elements only those subspaces which have eitherfinite dimension or finite codimension. This geometry is of denumerable rank.

The following definition regarding subgroups of Cor( ) is to be comparedwith Definition 1.4.3 for subgroups of Aut( ). It uses the action 7 I ofCor( ) introduced in Remark 1.3.2.

1.4 Subgeometries and truncations 19

Definition 1.4.12 Let A be a group of auto-correlations of . The absoluteof with respect to A is the incidence system A = (XA, A, A) over J ,where

(1) the set J is the collection of all A-orbits K on I for which there areinvariant flags of type K;

(2) the set XA consists of all minimal (which is understood to imply non-empty) A-invariant flags of ;

(3) the relation A on XA is determined by F A G if and only if F G is aflag of ;

(4) the function A : XA J is the map assigning to a minimal A-invariantflag F the set of A-orbits in (F ).

Suppose that A is a subgroup of Aut( ). Let J be the set of all j Ifor which there is an element of type j fixed by A. Replacing singletons oftypes by the actual type, we can view the absolute A as an incidence systemover J . After a similar replacement of singletons by their elements in XA, theabsolute A will coincide with the fixed subsystem of with respect to A.

Several important geometries to be met later on in this book, arise asabsolutes with respect to a small group A of correlations of some biggergeometry . They often have interesting simple groups of automorphismswhich are subgroups of Aut( ) centralizing A.

Example 1.4.13 Let V be a vector space of finite dimension n over a fieldF. A bilinear form on V is a map f : V V F such that, for each v V ,both maps

f(v, ) : V F, f(, v) : V Fare linear. A bilinear form f is called symmetric if, for all v, w V , we havef(v, w) = f(w, v).

The radical of f is the following linear subspace of V .

V = {x V | f(x, y) = 0 for all y V } (1.1)

It is also denoted Rad(f).Fix a basis 1, . . . , n of V . A typical example of a symmetric bilinear

form on V is f0(x, y) =n

i=1 xiyi where x =

i xii, y =

i yii with xi,yi F.

For x V , we denote by x the linear subspace of V consisting of ally V with f(x, y) = 0. It is either a hyperplane, that is, a subspace of Vof codimension 1, or all of V . If x is a hyperplane for each nonzero vector inV , the form f is called nondegenerate. Observe that f is nondegenerate ifand only if Rad(f) = {0}. The example f0 is nondegenerate. For an arbitrarysubset X of V , set X =

xX x

. Clearly, V = Rad(f).Suppose now that f is nondegenerate. If X is a k-dimensional subspace of

V , then X is an (n k)-dimensional subspace and (X) = X . Hence, the

20 1. Geometries

map : X 7 X is an auto-correlation of PG(V ) (the projective geometry ofExample 1.4.9) of order two; in other words, is a polarity. We consider theabsolute of PG(V ) with respect to A = . The A-orbit of a subspaceX of Vof dimension k consists of X and X. As dim (X) = n k and has order2, we may interchange X and X if needed, so as to force dim (X) n/2.Now X X, where is the incidence relation of PG(V ), means X X,which is equivalent to X being singular, that is, f(x, y) = 0 for all x, y X .Therefore, by mapping the element {X,X} of the absolute to the leastdimensional member of this unordered pair, we can identify the absolute ofPG(V ) with respect to A with the set of all singular subspaces of V of affinedimension at most n/2. Incidence and type are as in PG(V ): the former isgiven by symmetrized inclusion, whereas (X) = dim (X). This geometry isa polar geometry, which is the subject of Chapters 710.

1.5 Residues

The concept of a residue is a cornerstone of the theory of geometries. In thephysical view, where elements are sets of points, this concept arises in differentways, such as subspaces (for the elements contained in a given subspace) andquotient spaces (for the elements containing a given subspace). In the studyof polyhedra it appears as the vertex figure, that is, the collection of edgesand faces containing a vertex together with their incidences. Residues unifyand clarify these classical concepts. As usual, I will be a fixed set of typesand = (X, , ) an incidence system over I .

Definition 1.5.1 Let F be a flag of . The residue of F in is the sub-system F of over I\(F ) on the point set F \F .

For the sake of brevity, residues of flags of are also referred to as residuesof . If F = {x} for some x X , we also write x instead of F .

Example 1.5.2 Consider the cube as a geometry over {vertex, edge, face}.The residue of a face of the cube is a quadrangle. A little more thoughtyields that the residue of a vertex is a triangle over {edge, face}. The residueof an edge is a polygon with two vertices and two edges, which we call adigon. Other residues are less interesting. The residue of the empty flag isthe full cube, the residue of a chamber is empty. The residue of a rank twoflag consists of exactly two elements: the geometry is thin.

Similarly, consider the icosahedron (Figure 1.2) as a geometry over{vertex, edge, face}. This geometry is also thin. The residue of a face isa triangle. The residue of a vertex is a pentagon and the residue of an edge isa digon as defined in Example 1.5.2. In each polyhedron or tessellation, theresidue of an edge is a digon. Digons are very common indeed.

Proposition 1.5.3 Suppose that F is a flag of .

1.5 Residues 21

(i) A subset G of XF is a flag of F if and only if F G is a flag of .(ii) If G is a flag of F , then (F )G = FG.(iii) If is a geometry, then its residue F is a geometry over I\(F ).

Proof.

(i). Let G be a subset of XF . Then G is a flag of F if and only if G is a flagof and G F , which in turn is equivalent to G F being a flag of .(ii). Suppose that G is a flag of F . By (i), G F is a flag of . Now y is anelement of (F )G if and only if y (F \F )(G\G) = (F G)\(FG), thatis, y XFG. This proves (XF )G = XFG. As incidence and the type mapon (XF )G are both obtained by repeated restrictions, we find (F )G = FG.

(iii). This is an immediate consequence of (ii). tu

For a subset J of I and a flag F of , the notation JF can be interpretedas (J )F only if (F ) J and as J(F ) only if (F ) I\J . So confusionbetween the residue of F in the J-truncation of and the J-truncation ofthe residue of F in due to absence of brackets will not arise.

point line

Fig. 1.15. Polygons viewed as geometries

Example 1.5.4 A face-to-face tessellation of E3 by cubes is a geometry overI = {vertex, edge, face, cell}. The residue of a cell is a cube. The residueof a vertex consists of 6 edges, 12 faces, 8 cells, and is in fact an octahedron.

Example 1.5.5 Consider E3 as a geometry of rank three with points, lines,and planes. The residue of a plane is a plane, the real affine plane (of Example1.1.2). In the residue of a point, any two lines are incident with a unique planeand any two planes are incident with a unique line. These are features of thereal projective plane, that is PG(R3) (of Example 1.4.9). The residue ofa line consists of all points and all planes on that line. Every such point isincident with every such plane. This looks much like the digon structure inthat the incidence graph of this residue is a complete bipartite graph:each element is incident with each element of the other type. For this reasonit is called a generalized digon; it will be properly introduced in Definition2.1.1. We will write Km,n for the complete bipartite graph whose parts havesizes m and n.

22 1. Geometries

Example 1.5.6 Let V be a finite-dimensional vector space over the divisionring D and let PG(V ) be as in Example 1.4.9. Assume that U and W areincident subspaces of V with dim (W ) = dim (U) + 3. For every flag F ofPG(V ) maximal with respect to containing U and W but no subspaces ofdimension dim (U) + 1 or dim (U) + 2, the residue of F is PG(W/U), whichis isomorphic to the projective plane PG( D 3).

Also, if L is a line of PG(V ), so that dim (L) = 2, then the residue ofevery flag maximal with respect to containing L but not a point, is a set ofprojective points, which is identified with L in classical projective geometry.

1.6 Connectedness

This section deals with connectivity, the extent to which elements of a geom-etry can be joined by chains of incident elements. Again I is a set of types.

Definition 1.6.1 Let be a graph. If p and q are vertices of , then a pathof length n from p to q is a sequence p = x0, x1, x2, . . . , xn = q of vertices of such that {xi, xi+1} is an edge of for i = 0, . . . , n 1. We usually write

p = x0, x1, x2, . . . , xn = q

to indicate the path. The minimal length of a path from p to q is called thedistance between p and q and denoted by d(p, q), or just d(p, q) if it isclear in which graph the distance is measured. If there is no such path, wesay that the distance between p and q is . If d(p, q) = n, we also say thatp is at distance n from q. A path from p to q of length d(p, q) is called aminimal path or a geodesic from p to q in .

If we write p q whenever there exists some path (always of finite length)from p to q, then is clearly an equivalence relation on the vertex set of .The classes of are the connected components of . A graph is calledconnected if it has exactly one connected component, i.e., if the vertex setis non-empty and every pair of vertices are joined by a path.

Let = (X, , ) be an incidence system over I . Then is said to beconnected if its incidence graph is. A path in the incidence graph is calleda chain in . If, for J I , all of its vertices, except possibly the endpoints,have types in J , then it is called a J-chain.

If is a firm incidence system of rank one, then its incidence graph hasat least two vertices and no edges, so it is not connected. However, we will bemostly concerned with firm geometries of rank at least two that are connected.It is easy to draw pictures of connected geometries with disconnected residuesof rank two. Thus, connectedness leads to new restrictions on the geometriesof our interest.

1.6 Connectedness 23

Definition 1.6.2 An incidence system over I is called residually con-nected if each residue of of rank at least two is connected.

An incidence system over I is itself the residue of the empty flag. So, if|I | 2 and it is residually connected, then it is connected.

Lemma 1.6.3 Let I be finite and suppose that is a residually connected in-cidence system over I. If i, j are distinct elements of I and p, q two elementsof , then there exists an {i, j}-chain from p to q.

Proof. We proceed by induction on the rank r of = (X, , ). For r = 2 theproperty is obvious by connectedness of . Assume r 3. As is connected,there is a chain from p to q in , say

p = x0, x1, . . . , xn = q.

We want to find an {i, j}-chain from p to q. If p q, we are done, so we mayassume n > 1. Let a be the smallest index with 1 a < n such that xa is notof type i or j. We can assume that there is such an index, for otherwise theexisting chain is as required. Let k = (xa). The residue xa is a geometryover I\{k} all of whose residues of rank at least two are connected. Therefore,the induction hypothesis on r applies and gives an {i, j}-chain from xa1 toxa+1, inside xa . Figure 1.16 illustrates this argument.

p

k

q

i i i i i

j j j j j

i i i

Fig. 1.16. A chain from p to q

If we replace xa by this {i, j}-chain from xa1 to xa+1 in the original chain,we find a new chain. Applying the same procedure to each element of the chainwhose type is neither i nor j (there is one less of these in the new chain thanin the original chain), we eventually arrive at an {i, j}-chain from p to q. tu

Exercise 1.9.18 gives an example showing that the finiteness requirementon I in the lemma above is needed.

24 1. Geometries

The incidence system ({a, b}, , ) over [3] with (a) = 1 and (b) = 2, inwhich all (three) pairs of elements are incident, is residually connected butnot a geometry.

Lemma 1.6.4 A residually connected incidence system is a geometry if noflag of corank one is maximal.

Proof. Let F be a maximal flag of a residually connected incidence system over I (it exists by a straightforward application of Zorns Lemma). Weneed to show that F is a chamber. Assume it is not. By the hypotheses,|I\(F )| 2, where is the type map of . Now Definition 1.6.2 (recallthat the empty residue is not connected) yields the existence of an elementx of F , contradicting maximality of F . So, F must be a chamber. tu

For the case where I is finite, a weaker sufficient condition than the oneof Lemma 1.6.4 is given in Exercise 1.9.17.

Theorem 1.6.5 Suppose that is a residually connected incidence systemover I in which no flag of corank one is maximal. For each flag F of , theresidue F is a residually connected geometry.

Proof. Let F be a flag of . By Lemma 1.6.4, is a geometry and, byProposition 1.5.3(iii), F is a geometry over I\(F ), where is the type mapof .

Let G be a flag of F with |(I\(F ))\(G)| 2. As is residuallyconnected, Proposition 1.5.3(ii) gives that the residue (F )G = FG isconnected. Hence, F is residually connected. tu

Corollary 1.6.6 Let be a geometry over I and assume that I is finite.Then is residually connected if and only if, for all distinct i, j in I andflags F of having no elements of type i or j, the {i, j}-truncation {i,j}Fis connected.

Proof. The if part is obvious. As for the only if part, suppose that isresidually connected. By Theorem 1.6.5, the properties of carry over toF , so we can restrict ourselves to proving the result for F = . This meansthat it suffices to show that, for every two distinct i, j in I , the subgraph ofthe incidence graph of induced on Xi Xj is connected. For rk ( ) 1this is trivial and for rk ( ) 2 it is a consequence of Lemma 1.6.3. tu

To end this section, we single out the geometries of our interest.

Definition 1.6.7 A geometry over I is called an I-geometry if it is firmand residually-connected.

1.7 Permutation groups 25

1.7 Permutation groups

In previous sections, projective spaces, affine spaces, regular polytopes, andregular tilings were associated with geometries having a lot of symmetry,that is, large groups of automorphisms. In this section, we explore how largea group of automorphisms of a geometry need be to fully describe the ge-ometry in terms of the group. In fact, the main theorems are variations ofthe elementary but very basic Theorem 1.7.5 below which describes the corre-spondence between subgroups and transitive representations of a given group.

Throughout this section, G denotes a group and X a set. The groupSym(X) of all permutations of X was introduced in Remark 1.3.2. We willlet permutations act on the left. So, if Sym(X) and x X , then (x)denotes the image of x under . For n N, we often write Symn instead ofSym([n]). According to Definition 1.3.5, a group is a permutation group onX if it is a subgroup of Sym(X). We use the following slightly more generalnotion.

Definition 1.7.1 A representation of G in X is a group homomorphism : G Sym(X). In this case, X is referred to as a G-set. Such a represen-tation is called faithful if is injective.

Let x X . The set {(g)x | g G} is called the G-orbit of x in X(compare Definition 1.3.5). The representation is called transitive if X is asingle G-orbit. The stabilizer of x in G is the subgroup {g G | (g)x = x}of G; it is denoted by Gx.

As usual, we employ some abbreviations for the sake of readability. Insteadof a permutation representation of G on X , we often speak of an action ofG on X , and instead of (g)x, where g G, x X , we also write gx if therepresentation is clear from the context. Similarly, for the orbit (G)x ofx X we often write Gx. Observe also that is involved in the definition ofstabilizer but not visible in the corresponding notation.

IfH is a subgroup ofG, there is a standard way of constructing a transitiverepresentation of G on the space G/H = {aH | a G} of left cosets of H inG.

Definition 1.7.2 Let H be a subgroup of G. The map : G Sym(G/H)given by (g)aH = gaH for all g, a G is called the representation of Gover H .

It is readily seen that this map is a transitive permutation representa-tion. Its kernel is

aG aHa

1, the biggest normal subgroup of G containedin H .

26 1. Geometries

For H = 1, this representation is faithful. Therefore, G can be viewedas a subgroup of Sym(G) by means of the embedding sending g G to leftmultiplication by g on G.

Any transitive representation can be described as a representation over asubgroup. To be more precise, we need the notion of equivalence.

Definition 1.7.3 Let X be a set. Two representations : G Sym(X)and : G Sym(X ) are said to be equivalent if there is a bijection : X X such that (g)1 = (g) whenever g G. In this case, wealso say that X and X are isomorphic G-sets.

For instance, any two sets of the same size n have natural actions of Symnand as such are isomorphic Symn-sets.

Example 1.7.4 If is the representation of a group G over a subgroup Hand a G, then the stabilizer in G of aH is aHa1. Now is equivalent to therepresentation of G over aHa1 by means of the map : G/H G/(aHa1)given by (yH) = ya1(aHa1). As a consequence, is determined up toequivalence by the conjugacy class of subgroups of G to which H belongs.

Theorem 1.7.5 Let : G Sym(X) be a transitive representation. Forx X, the representation of G over Gx is equivalent to .

Proof. Take : X G/Gx to be the map given by

(y) = gGx if g G satisfies y = (g)x.

Note that is well defined. First, as X is a single G-orbit, for each y X ,there is g G with y = (g)x. Second, if g and g are elements of G, withy = (g)x and y = (g)x, then (g1g)x = x, so g1g Gx, whencegGx = g

Gx.Also is a bijection. For, if y, y X satisfy (y) = (y), then gGx =

gGx for g, g G with (g)x = y and (g)x = y. Taking h, h Gx

with gh = gh, we find y = (g)x = (gh)x = (gh)x = (g)x = y,proving that is injective indeed. Clearly, is also a surjection, and so evena bijection.

Now write for the representation of G over Gx, and let g G andz X . Then, with h G such that z = (h)x, we find

(g)(z) = ((gh)x) = ghGx = (g)(hGx) =

(g)(z),

and so is indeed an equivalence between the G-sets X and G/Gx. tu

1.7 Permutation groups 27

Remark 1.7.6 Here is a more general and more sophisticated formulation ofthe theorem that incorporates intransitive actions. Let X be a G-set. Denoteby X/G the collection of orbits of G on X (compare again Definition 1.3.5).So, if x X , then Gx belongs to X/G. Let t : X/G X be a transver-sal, i.e., a map satisfying Gt(y) = y for y X/G. Then there is a G-setisomorphism

yX/G

G/Gt(y) X

given by gGt(y) 7 gt(y) (g G, y X/G). In the same vein, the variationsof Theorem 1.7.5 for group actions on graphs and geometries obtained in thissection and the next one can be formulated more generally.

Example 1.7.7 From a given representation : G Sym(X), several oth-ers can be constructed. The homomorphism (2) : G Sym

(X2

)given by

(2)(g){x, y} = {(g)x, (g)y} (x, y X, x 6= y)

is an example. The constructed representations are often clear once the un-derlying set (which is

(X2

)in the above example) is specified. We then refer

to the representation as the action of G on this set induced by .

Example 1.7.8 We study the subgroups of Sym4. The definition of Sym4 asthe group of all permutations of the set [4] gives the following non-exhaustivelist of transitive Sym4-sets with indicated stabilizers.

(1) The set [4], with stabilizer isomorphic to Sym3; example: (1, 2), (2, 3)is the stabilizer of 4.

(2) The set([4]2

)(of size six) of all 2-subsets of [4], with stabilizer isomorphic

to C2 C2; example: (1, 2), (3, 4) is the stabilizer of {1, 2} (and also of{3, 4}).

(3) The set of ordered pairs of distinct points of [4], with stabilizer isomorphicto C2; example: (3, 4) is the stabilizer of the ordered pair (1, 2) [4][4].

(4) The set of partitions of [4] into two subsets of size two each, with sta-bilizer isomorphic to Dih8, the dihedral group of order eight; example:(1, 3), (1, 2, 3, 4) is the stabilizer of {{1, 3}, {2, 4}}.

It is also known that Sym4 is the group of all rotations of the cube (see alsoExample 1.7.15). To see this, label the vertices of the cube as in Figure 1.17.

There are four pairs of opposite vertices, namely 1 = {a, a}, 2 = {b, b},3 = {c, c}, and 4 = {d, d}. The element z := (a, a)(b, b)(c, c)(d, d) isinduced by the homothety id on the Euclidean vector space R3 in whichthe cube embeds (with center of gravity at the origin). It is not a rotation.Each element of Sym4 determines two automorphisms of the graph in Figure1.17, one of which is obtained from the other by multiplication by z. Inparticular, a unique automorphism corresponding to the element of Sym4 isa rotation of the cube. For instance, (3, 4) corresponds to the automorphism

28 1. Geometries

a b

cd

b a

dc

Fig. 1.17. The cube as a graph

(c, d)(c, d) of the graph which is induced by a reflection of R3 and so maps to(c, d)(c, d)z = (a, a)(b, b)(c, d)(c, d). This gives a faitfhul action of Sym4on the cube.

From this description we derive the following non-exhaustive list of tran-sitive Sym4-sets.

(5) The set of two tetrahedra in the cube, with stabilizer isomorphic to Alt4.(6) The set of faces, with stabilizer C4.(7) The vertex set, with stabilizer C3.(8) The set of flags of type {vertex, edge}, with stabilizer the trivial group.

So far, we have found eight conjugacy classes of subgroups of Sym4. Thereare two more classes of proper subgroups, with representatives (1, 2)(3, 4)(isomorphic to C2) and (1, 2)(3, 4), (1, 3)(2, 4) (isomorphic to C2 C2), re-spectively. In particular, there are two conjugacy classes of groups isomorphicto C2 in Sym4. Similarly for C2 C2 instead of C2.

The nontrivial proper subgroups of Sym4 obey the pattern displayed byFigure 1.18. Each box represents a conjugacy class of subgroups whose iso-morphism type is as inscribed. The cardinality of the conjugacy class can befound in the southwest corner of the box. Inclusions are as indicated by thelabelled connections between boxes: the group at the bottom is a maximalsubgroup of the one at the top (up to conjugacy). For instance, each memberof the class of Sym3 contains three members of the class of C2 of size six andeach of the latter is contained in two subgroups isomorphic to Sym3.

As a consequence, Sym4 has eleven inequivalent transitive representations.

We are ready for variations of Theorem 1.7.5 to structures whose under-lying set admits a representation. The group of the representation should acton the structure as a group of automorphisms.

1.7 Permutation groups 29

6

Dih

1

1

11

13

211 1

1

34

Alt3

33x2 2

22

1x2 2

3Sym

C C C C C

C C C

1

4

4 84

3

1

2

1 1

3

4

13

1

1

1

Fig. 1.18. The subgroup pattern of Sym4

Definition 1.7.9 If denotes a structure (e.g., a vector space) and Aut()the group of all automorphisms of , then we say that is a representationof G on if it is a group homomorphism : G Aut().

If is a vector space, this means that is a so-called linear representation.If is just a set, then Definition 1.7.1 coincides with Definition 1.7.9.

As before, we also speak of a G-action on when we mean a representa-tion in .

In order to state the variations of Theorem 1.7.5 for other structures, weneed the notion of equivalence of representations of these structures.

Definition 1.7.10 Two representations , of G on structures , (ofthe same kind), respectively, are called equivalent if there is an isomor-phism : establishing equivalence between the associated ordinaryrepresentations : G Sym() and : G Sym().

Here, Sym() is the symmetric group on the natural set underlying .For instance, the set of vectors if is a vector space.

We now focus on graph structures.

Definition 1.7.11 Suppose that is a graph and that G acts on . Arepresentation : G Aut() is called edge transitive if both and theinduced action of on the set of edges of are transitive.

Remark 1.7.12 We describe how to translate the data of a graph =(X,) with sufficient symmetry, namely a transitive representation : GAut(), into group information.

By Theorem 1.7.5, taking x X and setting H = Gx, we may assumewithout loss of generality that X = G/H and (g)aH = gaH (a, g G).It remains to describe in terms of the groups G and H . For a, b G, we

30 1. Geometries

have aH bH if and only if H a1bH , so that it suffices to identify theneighbors of the vertex H . To this end write K = {g G | H gH}. Clearly,K = hK = Kh for every h H , so K is a union of double H-cosets. Hencea graph on which G acts transitively is determined by the knowledge of astabilizer H and a union K of double cosets of H in G.

As is a graph, H rH implies rH H , whence, by application of(r1), also H r1H . This shows that r1 K, so K = K1. We nowhave a full description of terms of G, H , and K under the assumption thatG is edge transitive; aH bH if and only if a1b K (for a, b G).

Reversing the above, we start with a group G, a subgroup H and a unionK of double cosets ofH in G withK = K1. It is straightforward that (X,),where X = G/H and is the relation on X given by aH bH if and only ifa1b K (a, b G), is a graph admitting a transitive representation. Thisgraph has a vertex whose stabilizer is H and whose neighbors are the verticesaH for a K.

Definition 1.7.13 Let H be a subgroup of G and M a subset of G. Thegraph with vertex setG/H and edges the unordered pairs {aH, bH} (a, b G)with a1b kM (HkH Hk1H) is called the coset graph on G/Hdetermined by M . It is denoted (G,H,M).

The group G acts transitively on such a graph (G,H,M) by left multi-plication on the vertex set. We focus on the simplest case, whereK of Remark1.7.12 consists of a single double coset, so we can take M = {r} for a singler G with r1 HrH . This means that, if aH is adjacent to H for somea G, then aH = brH for some b H . In other words, H , the stabilizer inG of the vertex H in X , is transitive on the set of the vertices adjacent toH . So G is edge transitive on the graph (G,H, {r}).

Proposition 1.7.14 If : G Aut(X,) is an edge-transitive representa-tion of G on a graph (X,), then, for x X, the representation of G on thecoset graph (G,Gx, {r}), where r G satisfies x (r)x, is equivalent to.

Proof. Fix x X and take : G/Gx X to be the map sending gGxto gx. Then establishes the required isomorphism. tu

Example 1.7.15 Take G = Sym4 and H = (1, 2, 3). As we have seen inExample 1.7.8(7), the set G/H corresponds, as a G-set, to the vertex setof the cube, on which G acts. The graph of the cube is equivalent tothe coset graph (G,H, {(1, 4)}). Each vertex of has the same numberof vertices adjacent to it; such a graph is called regular and the numberis called the valency of . Let us verify that the valency of this graph isthree (as it should be). The valency is the number of H-cosets in H(1, 4)H .

1.7 Permutation groups 31

As H (1, 4)H(1, 4)1 = (1, 2, 3) (4, 2, 3) = 1, we have |H(1, 4)H | =|H(1, 4)H(1, 4)1| = |H | |H |/|H (1, 4)H(1, 4)1| = 9, and so the valencyof equals 9/|H | = 3.

Example 1.7.16 Let n N, n 2, and let G = Symn act on [n]. SinceG acts transitively on

([n]2

)(see Example 1.7.8), the only graph structures

on [n] preserved by G are the empty graph (with empty edge set) and the

complete graph (with edge set([n]2

)). Next consider graphs with vertex set

V =([n]2

). It is readily seen that G is transitive on the collection E of

ordered pairs (x, y) V V with = |x y| taking values in {0, 1, 2}. As|x y| = |yx|, the relations E are symmetric, that is, (x, y) E implies(y, x) E. Now E2 is the diagonal, {(x, x) | x V }. The other two G-orbits, E0 and E1, lead to graphs that are each others complements (thecomplement of a graph (V,E) is the graph (V,

(V2

)\E)). The equivalent coset

graphs are (G,H, {(2, 3)}) and (G,H, {(1, 3)(2, 4)}), respectively, whereH = (1, 2) Sym({3, . . . , n}). The distribution diagram of (V,E1) is

1 2n42n4

1(n2)(n3)/2

n3

4

2n8n2

Fig. 1.19. The distribution diagram of Symn acting on`[n]2

This diagram gives a summary of the division of the vertex set into H-orbits.The three H