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
13.
The Sylow Theorems
13.1.
p-Groups
13.1.1.
Definition and Examples
13.1.2.
Properties of p-Groups
13.1.3.
Centers of p-Groups
13.1.4.
Maximal p-Subgroups
13.2.
Sylow p-Subgroups
13.2.1.
Definition and Examples
13.2.2.
Maximal p-Subgroups
13.3.
The First Sylow Theorem
13.3.1.
Statement and Proof
13.3.2.
Existence of Sylow p-Subgroups
13.3.3.
Cauchy's Theorem
13.4.
The Second Sylow Theorem
13.4.1.
Statement and Proof
13.4.2.
Conjugacy of Sylow p-Subgroups
13.4.3.
Normalizer Conditions
13.5.
The Third Sylow Theorem
13.5.1.
Statement and Proof
13.5.2.
Number of Sylow p-Subgroups
13.5.3.
Congruence Conditions
13.6.
Applications of the Sylow Theorems
13.6.1.
Proving Non-simplicity
13.6.2.
Classifying Groups of Small Order
13.6.3.
Structure Theorems
13.6.4.
Counting Arguments
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14. Direct Products and Sums