Groups and Symmetry: Theory and Applications

Preface
Contents
About the Author
Notations
1 Group Actions on Sets
	1.1 Group Actions and G-Sets
	1.2 Orbits and Stabilizers
	1.3 Transitive G-Sets
	1.4 Primitive G-Sets
	1.5 Applications of Burnside's Lemma
	1.6 Pólya's Enumeration
	1.7 Worked-Out Problems
	1.8 Supplementary Exercises
2 Sylow's Theorems and Applications
	2.1 p-Groups
	2.2 Sylow's Theorems
	2.3 Sylow p-Subgroups of Specific Groups
	2.4 Worked-Out Problems
	2.5 Supplementary Exercises
3 Simple Groups
	3.1 Simple Groups and Examples
	3.2 Non-simplicity Tests
	3.3 The Simplicity of Alternating Groups
	3.4 The Simplicity of the Projective Special Linear Groups
	3.5 Worked-Out Problems
	3.6 Supplementary Exercises
4 Product of Groups
	4.1 External Direct Product
	4.2 Internal Direct Product
	4.3 Semidirect Product
	4.4 Finite Abelian Groups
	4.5 Worked-Out Problems
	4.6 Supplementary Exercises
5 Normal Series
	5.1 Jordan–Hölder Theorem
	5.2 Solvable Groups
	5.3 Nilpotent Groups
	5.4 Worked-Out Problems
	5.5 Supplementary Exercises
6 Free Groups and Presentations
	6.1 Categories and Free Objects
	6.2 Free Groups
	6.3 Generators and Relations
	6.4 Coxeter Groups
	6.5 Worked-Out Problems
	6.6 Supplementary Exercises
7 Symmetry Groups of Geometric Objects
	7.1 Isometries of Euclidean Space
	7.2 Isometries of mathbbR3
	7.3 Isometries of mathbbR2
	7.4 Finite Rotation Groups
	7.5 Worked-Out Problems
	7.6 Supplementary Exercises
8 Platonic Solids
	8.1 Platonic Solids
	8.2 Classification of Finite Groups of Rotations in mathbbR3
	8.3 Worked-Out Problems
	8.4 Supplementary Exercises
9 Frieze and Wallpaper Symmetry Groups
	9.1 Point Groups and Rosette Groups
	9.2 Frieze Groups
	9.3 Wallpaper Groups
	9.4 Worked-Out Problems
	9.5 Supplementary Exercises
10 Representations and Characters of Groups
	10.1 Modules
	10.2 Group Representations
	10.3 Reducible and Irreducible Representation
	10.4 Characters
	10.5 Algebraic Numbers and Algebraic Integers
	10.6 Complex Characters
	10.7 Tensor Products and Induced Characters
	10.8 Representations of Abelian Groups
	10.9 Burnside's paqb Theorem
	10.10 Worked-Out Problems
	10.11 Supplementary Exercises
Appendix  References
Index