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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
Structure of Finite Abelian Groups
The Fundamental Theorem of Finite Abelian Groups
Statement of the Theorem
Invariant Factor Form
Elementary Divisor Form
Primary Decomposition
Classification Results
Uniqueness of Decomposition
Counting Abelian Groups
Isomorphism Classes
Applications
Structure of (ℤ/nℤ)*
Solving Equations in Abelian Groups
Lattice of Subgroups
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14. Direct Products and Sums
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16. Solvable and Nilpotent Groups