This textbook provides a readable account of the examples and fundamental results of groups from a theoretical and geometrical point of view. This is the second book of the set of two books on groups theory. Topics on linear transformation and linear groups, group actions on sets, Sylow’s theorem, simple groups, products of groups, normal series, free groups, platonic solids, Frieze and wallpaper symmetry groups and characters of groups have been discussed in depth. Covering all major topics, this book is targeted to advanced undergraduate students of mathematics with no prerequisite knowledge of the discussed topics. Each section ends with a set of worked-out problems and supplementary exercises to challenge the knowledge and ability of the reader.
Author(s): Bijan Davvaz
Publisher: Springer
Year: 2021
Language: English
Pages: 294
City: Singapore
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
