UsefulLinks
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
15.
Structure of Finite Abelian Groups
15.1.
The Fundamental Theorem of Finite Abelian Groups
15.1.1.
Statement of the Theorem
15.1.2.
Invariant Factor Form
15.1.3.
Elementary Divisor Form
15.1.4.
Primary Decomposition
15.2.
Classification Results
15.2.1.
Uniqueness of Decomposition
15.2.2.
Counting Abelian Groups
15.2.3.
Isomorphism Classes
15.3.
Applications
15.3.1.
Structure of (ℤ/nℤ)*
15.3.2.
Solving Equations in Abelian Groups
15.3.3.
Lattice of Subgroups
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14. Direct Products and Sums
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16. Solvable and Nilpotent Groups