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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
The Isomorphism Theorems
Group Isomorphisms
Definition and Examples
Criteria for Isomorphism
Isomorphism as Equivalence Relation
The First Isomorphism Theorem
Statement and Proof
Applications and Examples
Fundamental Homomorphism Theorem
The Second Isomorphism Theorem
Statement and Proof
Diamond Isomorphism Theorem
The Third Isomorphism Theorem
Statement and Proof
Correspondence Theorem Applications
The Lattice Isomorphism Theorem
Statement and Applications
Correspondence of Subgroups
Correspondence of Normal Subgroups
Automorphisms
Definition of Automorphism
Automorphism Group
Inner Automorphisms
Definition and Properties
Inn(G) as Normal Subgroup
Outer Automorphisms
Definition and Examples
Out(G) = Aut(G)/Inn(G)
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10. Group Homomorphisms
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12. Group Actions