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
10.
Group Homomorphisms
10.1.
Definition of a Homomorphism
10.1.1.
Structure Preservation
10.1.2.
Homomorphism Condition
10.2.
Examples of Homomorphisms
10.2.1.
Trivial Homomorphism
10.2.2.
Inclusion Homomorphisms
10.2.3.
Determinant Map
10.2.4.
Sign Map for Permutations
10.2.5.
Exponential Map
10.2.6.
Reduction Modulo n
10.3.
Kernel of a Homomorphism
10.3.1.
Definition and Properties
10.3.2.
Kernel as a Normal Subgroup
10.3.3.
Trivial Kernel and Injectivity
10.4.
Image of a Homomorphism
10.4.1.
Definition and Properties
10.4.2.
Image as a Subgroup
10.4.3.
Surjectivity and Image
10.5.
Properties of Homomorphisms
10.5.1.
Preservation of Identity
10.5.2.
Preservation of Inverses
10.5.3.
Preservation of Powers
10.5.4.
Preservation of Order Relations
10.6.
Types of Homomorphisms
10.6.1.
Monomorphisms (Injective)
10.6.2.
Epimorphisms (Surjective)
10.6.3.
Isomorphisms (Bijective)
10.6.4.
Endomorphisms
10.6.5.
Automorphisms
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9. Normal Subgroups and Quotient Groups
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11. The Isomorphism Theorems