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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
Order of Elements and Groups
Definition of Order of a Group
Definition of Order of an Element
Computing Orders
Order in Finite Groups
Order in Infinite Groups
Relationship between Element Order and Group Order
Properties of Element Orders
Order of Powers
Order of Inverses
Order in Abelian Groups
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3. Fundamental Examples of Groups
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5. Subgroups