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
6.
Cyclic Groups
6.1.
Definition of a Cyclic Group
6.2.
Generators of a Group
6.2.1.
Definition of a Generator
6.2.2.
Finding Generators
6.2.3.
Multiple Generators
6.3.
Properties of Cyclic Groups
6.3.1.
Structure of Cyclic Groups
6.3.2.
All Subgroups are Cyclic
6.3.3.
Abelian Property
6.4.
Fundamental Theorem of Cyclic Groups
6.4.1.
Statement and Proof
6.4.2.
Uniqueness of Subgroups
6.5.
Subgroups of Cyclic Groups
6.5.1.
Number and Structure of Subgroups
6.5.2.
Lattice of Subgroups
6.5.3.
Correspondence with Divisors
6.6.
Classification of Cyclic Groups
6.6.1.
Infinite Cyclic Groups
6.6.1.1.
Isomorphism to ℤ
6.6.2.
Finite Cyclic Groups
6.6.2.1.
Isomorphism to ℤ/nℤ
6.7.
Generators of Finite Cyclic Groups
6.7.1.
Counting Generators
6.7.2.
Euler's Totient Function
6.7.3.
Primitive Roots
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7. Permutation Groups