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
11.
The Isomorphism Theorems
11.1.
Group Isomorphisms
11.1.1.
Definition and Examples
11.1.2.
Criteria for Isomorphism
11.1.3.
Isomorphism as Equivalence Relation
11.2.
The First Isomorphism Theorem
11.2.1.
Statement and Proof
11.2.2.
Applications and Examples
11.2.3.
Fundamental Homomorphism Theorem
11.3.
The Second Isomorphism Theorem
11.3.1.
Statement and Proof
11.3.2.
Diamond Isomorphism Theorem
11.4.
The Third Isomorphism Theorem
11.4.1.
Statement and Proof
11.4.2.
Correspondence Theorem Applications
11.5.
The Lattice Isomorphism Theorem
11.5.1.
Statement and Applications
11.5.2.
Correspondence of Subgroups
11.5.3.
Correspondence of Normal Subgroups
11.6.
Automorphisms
11.6.1.
Definition of Automorphism
11.6.2.
Automorphism Group
11.6.3.
Inner Automorphisms
11.6.3.1.
Definition and Properties
11.6.3.2.
Inn(G) as Normal Subgroup
11.6.4.
Outer Automorphisms
11.6.4.1.
Definition and Examples
11.6.4.2.
Out(G) = Aut(G)/Inn(G)
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10. Group Homomorphisms
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12. Group Actions