This volume studies groups as algebraic structures encoding symmetry.
This volume studies groups as algebraic structures encoding symmetry. It develops structural, combinatorial, and computational aspects, along with generalizations such as group actions and extensions.
Part I. Foundations of Group Theory
Chapter 1. Groups
1.1 Definitions and examples 1.2 Subgroups 1.3 Cyclic groups 1.4 Permutation groups 1.5 Basic properties
Chapter 2. Homomorphisms
2.1 Group homomorphisms 2.2 Kernels and images 2.3 Isomorphism theorems 2.4 Automorphisms 2.5 Examples
Chapter 3. Cosets and Lagrange’s Theorem
3.1 Cosets 3.2 Index of a subgroup 3.3 Lagrange’s theorem 3.4 Consequences 3.5 Applications
Part II. Structure Theory
Chapter 4. Normal Subgroups and Quotients
4.1 Normal subgroups 4.2 Quotient groups 4.3 Correspondence theorem 4.4 Examples 4.5 Applications
Chapter 5. Direct and Semidirect Products
5.1 Direct products 5.2 Internal vs external constructions 5.3 Semidirect products 5.4 Applications 5.5 Examples
Chapter 6. Group Actions
6.1 Actions on sets 6.2 Orbits and stabilizers 6.3 Orbit–stabilizer theorem 6.4 Applications 6.5 Examples
Part III. Finite Group Theory
Chapter 7. Sylow Theorems
7.1 p-groups 7.2 Sylow subgroups 7.3 Existence and conjugacy 7.4 Applications 7.5 Examples
Chapter 8. Classification Techniques
8.1 Solvable groups 8.2 Nilpotent groups 8.3 Simple groups 8.4 Composition series 8.5 Jordan–Hölder theorem
Chapter 9. Finite Simple Groups (Overview)
9.1 Definition 9.2 Examples 9.3 Classification theorem (overview) 9.4 Applications 9.5 Significance
Part IV. Infinite Groups
Chapter 10. Infinite Group Structures
10.1 Infinite cyclic groups 10.2 Free groups 10.3 Generators and relations 10.4 Presentations 10.5 Examples
Chapter 11. Growth and Geometry
11.1 Growth of groups 11.2 Cayley graphs 11.3 Geometric group theory (overview) 11.4 Hyperbolic groups (overview) 11.5 Applications
Chapter 12. Topological Groups (Overview)
12.1 Groups with topology 12.2 Continuity of operations 12.3 Examples 12.4 Haar measure (overview) 12.5 Applications
Part V. Representation Theory
Chapter 13. Group Representations
13.1 Linear representations 13.2 Modules over group algebras 13.3 Irreducible representations 13.4 Characters 13.5 Examples
Chapter 14. Character Theory
14.1 Character tables 14.2 Orthogonality relations 14.3 Applications 14.4 Induced representations 14.5 Examples
Chapter 15. Representation of Finite Groups
15.1 Maschke’s theorem 15.2 Decomposition 15.3 Applications 15.4 Examples 15.5 Connections
Part VI. Group Extensions and Cohomology
Chapter 16. Group Extensions
16.1 Extension problems 16.2 Split extensions 16.3 Classification 16.4 Applications 16.5 Examples
Chapter 17. Cohomology of Groups
17.1 Definitions 17.2 Low-dimensional cohomology 17.3 Extensions and cohomology 17.4 Applications 17.5 Examples
Chapter 18. Homological Methods
18.1 Resolutions 18.2 Derived functors 18.3 Applications 18.4 Examples 18.5 Connections
Part VII. Generalizations and Applications
Chapter 19. Group-Like Structures
19.1 Semigroups and monoids 19.2 Groupoids 19.3 Loop structures 19.4 Higher algebraic structures 19.5 Examples
Chapter 20. Applications of Group Theory
20.1 Symmetry in physics 20.2 Crystallography 20.3 Coding theory 20.4 Cryptography 20.5 Combinatorics
Chapter 21. Computational Group Theory
21.1 Algorithms for groups 21.2 Permutation group computation 21.3 Complexity issues 21.4 Software systems 21.5 Applications
Part VIII. Research Directions
Chapter 22. Advanced Topics
22.1 Algebraic groups (overview) 22.2 Lie groups connections 22.3 Geometric group theory 22.4 Infinite-dimensional groups 22.5 Emerging areas
Chapter 23. Open Problems
23.1 Classification challenges 23.2 Representation theory gaps 23.3 Algorithmic complexity 23.4 Structural questions 23.5 Future directions
Chapter 24. Historical and Conceptual Notes
24.1 Development of group theory 24.2 Key contributors 24.3 Evolution of ideas 24.4 Cross-disciplinary impact 24.5 Summary
Appendix
A. Common identities and theorems B. Standard group examples C. Proof techniques checklist D. Algorithm templates E. Cross-reference to other MSC branches
This volume develops group theory as the mathematics of symmetry. It connects algebra, geometry, and applications through structure and action.