Visual Group Theory: A Computer-Oriented Geometric Introduction (Springer Undergraduate Mathematics Series)

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