Table of contents

Group Theory from First Principles to Research Frontiers

A graduate path from axioms, structure, and representations to modern applications and open directions

Read each section in order. Every title can be opened as a TheoryTrace document.

• Cover
• Copyright
• How to read this book
• Introduction
• Chapter 1: Groups, Symmetry, and the Algebraic Viewpoint
• Chapter 2: Subgroups, Cyclic Groups, and Generators
• Chapter 3: Cosets, Lagrange’s Theorem, and Quotient Thinking
• Chapter 4: Homomorphisms, Isomorphisms, and Universal Properties
• Chapter 5: Normal Subgroups, Quotient Groups, and Extensions
• Chapter 6: Group Actions and the Orbit-Stabilizer Principle
• Chapter 7: Counting with Group Actions
• Chapter 8: Sylow Theory and the Structure of Finite Groups
• Chapter 9: Abelian Groups and Modules over the Integers
• Chapter 10: Free Groups, Presentations, and Generators and Relations
• Chapter 11: Commutators, Solvable Groups, and Nilpotent Groups
• Chapter 12: Composition Series and the Jordan-Hölder Theorem
• Chapter 13: Permutation Groups and Symmetric Groups
• Chapter 14: Linear Groups and Matrix Groups
• Chapter 15: Representation Theory of Finite Groups
• Chapter 16: Character Theory and Deeper Finite Group Methods
• Chapter 17: Lie Groups and Lie Algebras
• Chapter 18: Compact Lie Groups and Their Representations
• Chapter 19: Algebraic Groups and Groups over Fields
• Chapter 20: Topological, Profinite, and Locally Compact Groups
• Chapter 21: Geometric and Combinatorial Group Theory
• Chapter 22: Groups in Number Theory and Galois Theory
• Chapter 23: Groups in Geometry, Topology, and Physics
• Chapter 24: Computational Group Theory
• Chapter 25: The Classification of Finite Simple Groups
• Chapter 26: Current Research Frontiers in Group Theory
• Conclusion