Groups: A Path to Geometry

Following the same successful approach as Dr. Burn's previous book on number theory, this text consists of a carefully constructed sequence of questions that will enable the reader, through participation, to study all the group theory covered by a conventional first university course. An introduction to vector spaces, leading to the study of linear groups, and an introduction to complex numbers, leading to the study of Möbius transformations and stereographic projection, are also included. Quaternions and their relationships to 3-dimensional isometries are covered, and the climax of the book is a study of the crystallographic groups, with a complete analysis of these groups in two dimensions. Front Cover 1 Title 5 Copyright 6 Contents 7 Preface 13 1 Functions 15 Summary 21 Historical note 22 Answers 23 2 Permutations of a finite set 25 Summary 33 Historical note 33 Answers 34 3 Groups of permutations of R and C 37 Summary 47 Historical note 47 Answers 49 4 The Mobius group 54 Summary 64 Historical note 66 Answers 67 5 The regular solids 71 Summary 74 Historical note 74 Answers 75 6 Abstract groups 76 Summary 86 Historical note 86 Answers 88 7 Inversions of the Mobius plane and stereographic projection 91 Summary 98 Historical note 98 Answers 99 8 Equivalence relations 102 Summary 105 Historical note 105 Answers 106 9 Cosets 107 Summary 111 Historical note 112 Answers 113 10 Direct product 115 Summary 117 Historical note 117 Answers 118 11 Fields and vector spaces 119 Summary 124 Historical note 124 Answers 126 12 Linear transformations 128 Summary 130 Historical note 131 Answers 132 13 The general linear group GL(2, F) 133 Summary 137 Historical note 138 Answers 140 14 The vector space V_3(F) 142 Summary 146 Historical note 147 Answers 148 15 Eigenvectors and eigenvalues 150 Summary 157 Historical note 157 Answers 159 16 Homomorphisms 162 Summary 166 Historical note 166 Answers 167 17 Conjugacy 169 Summary 176 Historical note 176 Answers 177 18 Linear fractional groups 181 Summary 187 Historical note 188 Answers 189 19 Quaternions and rotations 192 Summary 196 Historical note 197 Answers 198 20 Affine groups 199 Summary 202 Historical note 202 Answers 204 21 Orthogonal groups 205 Summary 213 Historical note 214 Answers 215 22 Discrete groups fixing a line 219 Summary 223 Historical note 224 Answers 225 23 Wallpaper groups 227 Summary 239 Historical note 239 Answers 241 Bibliography 250 Index 253 https://archive.org/details/groupspathtogeom0000burn Group theory; Transformation groups; Geometry; Gruppentheorie; Algebra Groups This book follows the same successful approach as Dr Burn's previous book on number theory. It consists of a carefully constructed sequence of questions which will enable the reader, through his or her own participation, to generate all the group theory covered by a conventional first university course. An introduction to vector spaces, leading to the study of linear groups, and an introduction to complex numbers, leading to the study of Möbius transformations and stereographic projection, are also included. Quaternions and their relationship to three-dimensional isometries are covered, and the climax of the book is a study of crystallographic groups, with a complete analysis of these groups in two dimensions xii, 242 pages : 24 cm Includes bibliographical references (pages 236-237) and index