This textbook provides an introduction to group theory starting from the basics, relying on geometry to elucidate its various aspects. Groups naturally manifest as symmetries of geometric shapes, such as reflections and rotations. The book adopts this perspective to provide a straightforward, descriptive explanation, supported by examples and exercises in GAP, an open-source computer algebra system. It covers all of the key concepts of group theory, including homomorphisms, group operations, presentations, products of groups, and finite, abelian, and solvable groups. The topics include cyclic and symmetric groups, dihedral, orthogonal, and hyperbolic groups, as well as the significant notion of Cayley graphs. Self-contained and requiring little beyond high school mathematics, this book is aimed at undergraduate courses and features numerous exercises. It will also appeal to anyone interested in the geometric approach to group theory. Preface Contents 1 Introduction to Euclidean Geometry 1.1 Isometries 1.2 Figures and Permutations 1.3 The Structure of Isometries 1.4 Higher-Dimensional Spaces 2 Introduction to Groups 2.1 The Definition of a Group and the Dihedral Groups 2.2 The Order of a Group and Abelian Groups 2.3 Cyclic Groups 2.4 Properties of Groups 2.5 The Order of an Element 3 Subgroups and Homomorphisms 3.1 Subgroups 3.2 Cosets and Lagrange's Theorem 3.3 Homomorphisms 3.4 Normal Subgroups 3.5 Translations 4 Group Operations 4.1 The Symmetric Group 4.2 Operations of Groups on Sets 4.3 Conjugation 4.4 The Orbit-Stabilizer Theorem and the Class Equation 4.5 Cayley Graphs 4.6 A Decomposition of the Plane 5 Group Presentations 5.1 Group Presentations 5.2 Free Groups 5.3 Tietze Transformations and Decidability 6 Products of Groups 6.1 The Direct Product 6.2 The Free Product 6.3 The Semidirect Product 6.4 Discontinuous Groups and Translations 7 Finite Groups 7.1 An Example Exercises 7.2 The Sylow Theorems Exercises 7.3 Some Groups of Small Order Exercises 7.4 The Orthogonal Group Exercises 7.5 Regular Decompositions of the 2-Sphere Exercises 7.6 Counting Orbits Exercises 8 Abelian and Solvable Groups 8.1 Commutators Exercises 8.2 Abelian Groups Exercises 8.3 Solvable Groups Exercises 9 The Hyperbolic Plane 9.1 Axiomatic Geometry 9.2 Isometries in the Hyperbolic Plane Exercises 9.3 Decompositions of the Hyperbolic Plane Exercises 10 Hyperbolic Groups 10.1 Van Kampen Diagrams Exercises 10.2 Quasi-Isometries and the Švarc–Milnor Theorem Exercises 10.3 Isoperimetric Inequalities Exercises 10.4 Hyperbolic Groups Exercises 10.5 Combings Exercises A The Isometries of the Plane B Matrices C List of Symbols D Important Groups E Used GAP Commands F Hints for the Exercises G Notes on the Literature References Index
