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Mathematics
Group Theory
1. Introduction to Algebraic Structures
2. Definition of a Group
3. Fundamental Examples of Groups
4. Order of Elements and Groups
5. Subgroups
6. Cyclic Groups
7. Permutation Groups
8. Cosets and Lagrange's Theorem
9. Normal Subgroups and Quotient Groups
10. Group Homomorphisms
11. The Isomorphism Theorems
12. Group Actions
13. The Sylow Theorems
14. Direct Products and Sums
15. Structure of Finite Abelian Groups
16. Solvable and Nilpotent Groups
17. Composition Series and Jordan-Hölder Theorem
18. Free Groups and Presentations
19. Semidirect Products
20. Introduction to Representation Theory
21. Applications of Group Theory
20.
Introduction to Representation Theory
20.1.
Linear Representations
20.1.1.
Definition of Group Representation
20.1.2.
Matrix Representations
20.1.3.
Degree of Representation
20.1.4.
Equivalent Representations
20.2.
Examples of Representations
20.2.1.
Trivial Representation
20.2.2.
Regular Representation
20.2.3.
Permutation Representations
20.3.
Characters
20.3.1.
Definition of Character
20.3.2.
Character Tables
20.3.3.
Orthogonality Relations
20.4.
Irreducible Representations
20.4.1.
Definition and Examples
20.4.2.
Schur's Lemma
20.4.3.
Complete Reducibility
20.5.
Applications
20.5.1.
Burnside's Theorem
20.5.2.
Representation Theory of Finite Groups
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19. Semidirect Products
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21. Applications of Group Theory