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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
Semidirect Products
Definition and Construction
External Semidirect Product
Internal Semidirect Product
Action by Automorphisms
Examples and Properties
Dihedral Groups as Semidirect Products
Affine Groups
Holomorphs
Classification Results
Groups of Order pq
Extensions of Groups
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18. Free Groups and Presentations
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20. Introduction to Representation Theory