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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.

Cover1
Copyright2
How to read this book3
Introduction4
Chapter 1: Groups, Symmetry, and the Algebraic Viewpoint5
Chapter 2: Subgroups, Cyclic Groups, and Generators6
Chapter 3: Cosets, Lagrange’s Theorem, and Quotient Thinking7
Chapter 4: Homomorphisms, Isomorphisms, and Universal Properties8
Chapter 5: Normal Subgroups, Quotient Groups, and Extensions9
Chapter 6: Group Actions and the Orbit-Stabilizer Principle10
Chapter 7: Counting with Group Actions11
Chapter 8: Sylow Theory and the Structure of Finite Groups12
Chapter 9: Abelian Groups and Modules over the Integers13
Chapter 10: Free Groups, Presentations, and Generators and Relations14
Chapter 11: Commutators, Solvable Groups, and Nilpotent Groups15
Chapter 12: Composition Series and the Jordan-Hölder Theorem16
Chapter 13: Permutation Groups and Symmetric Groups17
Chapter 14: Linear Groups and Matrix Groups18
Chapter 15: Representation Theory of Finite Groups19
Chapter 16: Character Theory and Deeper Finite Group Methods20
Chapter 17: Lie Groups and Lie Algebras21
Chapter 18: Compact Lie Groups and Their Representations22
Chapter 19: Algebraic Groups and Groups over Fields23
Chapter 20: Topological, Profinite, and Locally Compact Groups24
Chapter 21: Geometric and Combinatorial Group Theory25
Chapter 22: Groups in Number Theory and Galois Theory26
Chapter 23: Groups in Geometry, Topology, and Physics27
Chapter 24: Computational Group Theory28
Chapter 25: The Classification of Finite Simple Groups29
Chapter 26: Current Research Frontiers in Group Theory30
Conclusion31

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