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
14.
Direct Products and Sums
14.1.
External Direct Product
14.1.1.
Definition and Construction
14.1.2.
Properties of External Direct Products
14.1.3.
Order of Elements in Direct Products
14.1.4.
Subgroups of Direct Products
14.2.
Internal Direct Product
14.2.1.
Definition and Conditions
14.2.2.
Recognition Theorem
14.2.3.
Relationship to External Direct Product
14.3.
Direct Sum of Abelian Groups
14.3.1.
Definition for Abelian Groups
14.3.2.
Relationship to Direct Product
14.3.3.
Infinite Direct Sums
14.4.
Decomposition Theorems
14.4.1.
Unique Factorization
14.4.2.
Indecomposable Groups
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13. The Sylow Theorems
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15. Structure of Finite Abelian Groups