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
12.
Group Actions
12.1.
Definition of a Group Action
12.1.1.
Left Actions
12.1.2.
Right Actions
12.1.3.
Faithful Actions
12.2.
Examples of Group Actions
12.2.1.
Left Regular Action
12.2.2.
Right Regular Action
12.2.3.
Action by Conjugation
12.2.4.
Action on Cosets
12.2.5.
Action on Subgroups
12.2.6.
Linear Actions
12.3.
Orbits
12.3.1.
Definition and Properties
12.3.2.
Orbit Decomposition
12.3.3.
Transitive Actions
12.4.
Stabilizers
12.4.1.
Definition and Properties
12.4.2.
Isotropy Subgroups
12.4.3.
Pointwise and Setwise Stabilizers
12.5.
The Orbit-Stabilizer Theorem
12.5.1.
Statement and Proof
12.5.2.
Applications and Examples
12.5.3.
Counting Orbits
12.6.
Fixed Points
12.6.1.
Definition and Examples
12.6.2.
Fixed Point Sets
12.6.3.
Free Actions
12.7.
Burnside's Lemma
12.7.1.
Statement and Proof
12.7.2.
Cauchy-Frobenius Lemma
12.7.3.
Applications to Counting
12.7.4.
Pólya Enumeration
12.8.
Conjugacy and Class Equation
12.8.1.
Conjugacy Classes
12.8.2.
Class Equation
12.8.3.
Applications to p-Groups
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11. The Isomorphism Theorems
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13. The Sylow Theorems