Miniquaternion Geometry: An Introduction to the Study of Projective Planes (Cambridge Tracts in Mathematics, Series Number 60)

This tract provides an introduction to four finite geometrical systems and to the theory of projective planes. Of the four geometries, one is based on a nine-element field and the other three can be constructed from the nine-element 'miniquaternion algebra', a simple system which has many though not all the properties of a field. The three systems based on the miniquaternion algebra have widely differing properties; none of them has the homogeneity of structure which characterizes geometry over a field. While these four geometries are the main subject of this book, many of the ideas developed are of much more general significance. The authors have assumed a knowledge of the simpler properties of groups, fields, matrices and transformations (mappings), such as is contained in a first course in abstract algebra. Development of the nine-element field and the miniquaternion system from a prescribed set of properties of the operations of addition and multiplication are covered in an introductory chapter. Exercises of varying difficulty are integrated with the text. Contents Preface Part I. Algebraic Background Chapter 1. Two Algebraic Systems with Nine Elements 1.1 Near-fields of order 9 1.2 The Galois field F of order 9 1.3 The miniquaternion system 2 1.4 The automorphism group of 2 1.5 The solution of equations in 2 Part Il. Field-Planes Chapter 2. Projective Planes 2.1 Elementary properties, subplanes 2.2 Collineations in a projective plane 2.3 The plane over an arbitrary field 2.4 The projectivities of Π(K) 2.5 Change of coordinates. Similarity 2.6 Conics and polarities in Π(K) 2.7 Projective correlations of Π(K) Chapter 3. Galois Planes of Orders 3 and 9 3.1 The construction of the plane Δ of order 3 3.2 The collineation group of Δ 3.3 The Galois plane Φ of order 9 3.4 Subplanes of Φ 3.5 Sets of mutually disjoint subplanes 3.6 Singer cycles in Φ 3.7. Conics in Φ 3.8 Hermitian sets in Φ Part III. Miniquaternion Planes Chapter 4. The Planes Ω and Ω^D 4.1 The construction of the plane Ω 4.2 Some collineations of Ω 4.3 The collineation group of Ω 4.4 Fano subplanes of Ω 4.5 Subplanes of order 3 in Ω 4.6 An oval and some constructions in Ω 4.7 The dual plane Ω^D Chapter 5. The Plane Ψ 5.1 The construction and the collineation group of Ψ 5.2 Equations of lines and the self-duality of Ψ 5.3 Some constructions in Ψ 5.4 Polarities in Ψ 5.5 A combinatorial analysis of Ψ 5.6 The derivation of Ψ from Φ 5.7 The subplanes of Ψ List of Special Symbols References Suggestions for Further Reading Index