Linear Geometries
In Example 1.4.9 we introduced the projective geometry PG(V) and in Example 1.4.10 the affine geometry AG(V) associated with a vector space V of finite dimension n. In Proposition 2.4.7 the geometry PG(V) was shown to have a linear Coxeter diagram A n−1,
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Linear Geometries
In Example 1.4.9 we introduced the projective geometry PG(V ) and in Example 1.4.10 the affine geometry AG(V ) associated with a vector space V of finite dimension n. In Proposition 2.4.7 the geometry PG(V ) was shown to have a linear Coxeter diagram An−1 , and in Proposition 2.4.10 the geometry AG(V ) was shown to belong to the linear diagram Afn . We now turn our attention to the more general class of all geometries with a linear diagram. The shadow spaces on 1 of our motivating examples PG(V ) and AG(V ) are linear line spaces (in the sense that any two points are on a unique line; cf. Definition 2.5.13), and we will restrict ourselves mostly to geometries with this property. Within this class there are combinatorial structures such as matroids and Steiner systems. In Sect. 5.1, we introduce affine space as an abstraction of a vector space, and in Sect. 5.2 we do the same for projective space. The point shadows of the geometries AG(V ) are examples of affine spaces and those of PG(V ) are examples of projective spaces. In Sect. 5.3 we make the connection between geometries with a linear diagram and matroids, whereas in Sect. 5.4 we show how to build a geometry with a linear diagram from a matroid. In Sect. 5.5 we devote attention to Steiner systems and in particular the unique Steiner system S(5, 8, 24) related to the Golay code. Finally, in Sect. 5.6, we study its automorphism group, which is a sporadic simple group, the Mathieu group on 24 letters. Although the last two sections are not needed for the remainder of the book, the treatment of this Mathieu group (as well as the other four Mathieu groups, which appear as groups of automorphisms of substructures in the Steiner system S(5, 8, 24)) shows how they appear naturally in diagram geometry. In Sect. 5.8, the notes to this chapter, we give a linear diagram for a geometry of each of the 26 sporadic finite simple groups, with references to the literature.
5.1 The Affine Space of a Vector Space Affine spaces, the subject of this section, are abstractions of vector spaces. We formulate the abstract definition of an affine space and show that the well-known affine F. Buekenhout, A.M. Cohen, Diagram Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 57, DOI 10.1007/978-3-642-34453-4_5, © Springer-Verlag Berlin Heidelberg 2013
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Linear Geometries
space associated with a vector space is an example. In Chap. 6, the abstract definition will be used for an axiomatic characterization of the affine space of a vector space. We state some facts without proofs because they do not go beyond elementary linear algebra. We take the scalars of a vector space to come from a division ring rather than a field. The main reason for this greater generality is that the class of affine spaces over a division ring will be characterized by a simple set of axioms and that the restriction to fields would not imply an essential simplification. For the duration of the section, let D be a division r
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