Here’s a list of 100 chapter titles for Group Theory, organized from beginner to advanced levels. These titles cover foundational concepts, applications, and advanced theoretical aspects of group theory in mathematics:
- Introduction to Group Theory: What is a Group?
- Basic Properties of Groups: Closure, Associativity, Identity, and Inverses
- Examples of Finite Groups: Cyclic Groups
- Examples of Infinite Groups: The Integers under Addition
- Understanding Group Tables (Cayley Tables)
- Subgroups: Definition and Examples
- The Order of a Group and the Order of an Element
- Cyclic Groups and Their Generators
- The Symmetric Group: Permutations and Notation
- The Alternating Group: Even and Odd Permutations
- Dihedral Groups: Symmetries of Regular Polygons
- The Klein Four-Group: A Non-Cyclic Abelian Group
- The Quaternion Group: A Non-Abelian Group
- Homomorphisms: Structure-Preserving Maps
- Isomorphisms: When Are Two Groups the Same?
- Automorphisms: Symmetries of a Group
- The Center of a Group: Commuting Elements
- The Centralizer and Normalizer of a Subgroup
- Cosets: Left and Right Cosets
- Lagrange’s Theorem: The Order of Subgroups
- Normal Subgroups: Definition and Examples
- Factor Groups (Quotient Groups): Constructing New Groups
- The First Isomorphism Theorem: Connecting Homomorphisms and Quotients
- Simple Groups: Groups with No Non-Trivial Normal Subgroups
- Direct Products of Groups: Combining Groups
- Abelian Groups: Commutative Groups
- Finite Abelian Groups: Structure and Classification
- The Fundamental Theorem of Finite Abelian Groups
- Group Actions: Groups Acting on Sets
- Orbits and Stabilizers in Group Actions
- The Class Equation: Counting Elements in Conjugacy Classes
- Conjugacy Classes: Elements Sharing the Same Structure
- The Sylow Theorems: Subgroups of Prime Power Order
- Applications of the Sylow Theorems
- Solvable Groups: Groups with Abelian Composition Factors
- Nilpotent Groups: A Special Class of Solvable Groups
- The Commutator Subgroup: Measuring Non-Abelianness
- The Derived Series: Building Solvable Groups
- The Lower Central Series: Building Nilpotent Groups
- Free Groups: Groups with No Relations
- Presentations of Groups: Generators and Relations
- The Word Problem in Group Theory
- Torsion Groups: Groups Where Every Element Has Finite Order
- Torsion-Free Groups: Groups with No Non-Trivial Finite-Order Elements
- Finitely Generated Groups: Groups with Finite Generating Sets
- The Burnside Problem: Groups of Finite Exponent
- The Krull-Schmidt Theorem: Uniqueness of Direct Product Decompositions
- The Jordan-Hölder Theorem: Uniqueness of Composition Series
- The Schreier Refinement Theorem: Refining Subgroup Series
- The Zassenhaus Lemma: A Tool for Refining Subgroup Series
- The Diamond Lemma: A Visual Tool for Subgroup Lattices
- The Lattice of Subgroups: Visualizing Subgroup Relationships
- The Frattini Subgroup: The Intersection of All Maximal Subgroups
- The Fitting Subgroup: The Largest Normal Nilpotent Subgroup
- The Socle of a Group: The Subgroup Generated by Minimal Normal Subgroups
- The Jacobson Radical of a Group: A Measure of Non-Simplicity
- The Schur-Zassenhaus Theorem: Splitting Extensions
- The Hall Theorems: Subgroups of Coprime Order
- The Feit-Thompson Theorem: Odd-Order Groups Are Solvable
- The Classification of Finite Simple Groups: An Overview
- Representation Theory: Groups as Linear Transformations
- Characters of Finite Groups: Traces of Representations
- The Character Table: Encoding Group Information
- Irreducible Representations: Building Blocks of Representations
- Maschke’s Theorem: Complete Reducibility of Representations
- Schur’s Lemma: A Tool for Irreducible Representations
- The Tensor Product of Representations
- The Dual Representation: Contragredient Representations
- Induced Representations: Building Representations from Subgroups
- Restriction of Representations: Reducing Representations to Subgroups
- Frobenius Reciprocity: Connecting Induction and Restriction
- The Peter-Weyl Theorem: Decomposing Group Algebras
- The Artin-Wedderburn Theorem: Structure of Group Algebras
- The Brauer-Nesbitt Theorem: Characterizing Representations
- The Mackey Decomposition Theorem: Analyzing Induced Representations
- The Mackey Irreducibility Criterion: Testing Induced Representations
- The Clifford Theory: Representations of Normal Subgroups
- The Schur Functor: Constructing New Representations
- The Schur-Weyl Duality: Connecting Symmetric and General Linear Groups
- The Weyl Character Formula: Computing Characters of Lie Groups
- The Harish-Chandra Homomorphism: Relating Lie Algebras and Representations
- The Borel-Weil-Bott Theorem: Constructing Representations via Geometry
- The Langlands Program: Connecting Group Theory and Number Theory
- The Tannaka-Krein Duality: Reconstructing Groups from Representations
- The Gelfand-Naimark Theorem: Representations of C*-Algebras
- The Kazhdan-Lusztig Theory: Representations of Hecke Algebras
- The Deligne-Lusztig Theory: Representations of Finite Groups of Lie Type
- The Geometric Satake Correspondence: Connecting Group Theory and Geometry
- The Local Langlands Correspondence: Connecting Group Theory and Automorphic Forms
- The Global Langlands Correspondence: A Grand Unified Theory
- Infinite Groups: Properties and Examples
- Profinite Groups: Limits of Finite Groups
- Pro-p Groups: Groups with Prime-Power Order
- Lie Groups: Groups with Smooth Manifold Structure
- Algebraic Groups: Groups Defined by Polynomial Equations
- Quantum Groups: Deformations of Classical Groups
- Braid Groups: Groups of Braids and Their Applications
- Coxeter Groups: Groups Generated by Reflections
- Hyperbolic Groups: Groups with Negative Curvature
- The Future of Group Theory: Open Problems and Research Directions
This progression ensures a comprehensive understanding of group theory, starting from basic concepts and gradually moving toward advanced mathematical theories and applications.