Group Theory

  1. Group Homomorphisms
    1. Definition of a Homomorphism
      1. Structure Preservation
        1. Homomorphism Condition
        2. Examples of Homomorphisms
          1. Trivial Homomorphism
            1. Inclusion Homomorphisms
              1. Determinant Map
                1. Sign Map for Permutations
                  1. Exponential Map
                    1. Reduction Modulo n
                    2. Kernel of a Homomorphism
                      1. Definition and Properties
                        1. Kernel as a Normal Subgroup
                          1. Trivial Kernel and Injectivity
                          2. Image of a Homomorphism
                            1. Definition and Properties
                              1. Image as a Subgroup
                                1. Surjectivity and Image
                                2. Properties of Homomorphisms
                                  1. Preservation of Identity
                                    1. Preservation of Inverses
                                      1. Preservation of Powers
                                        1. Preservation of Order Relations
                                        2. Types of Homomorphisms
                                          1. Monomorphisms (Injective)
                                            1. Epimorphisms (Surjective)
                                              1. Isomorphisms (Bijective)
                                                1. Endomorphisms
                                                  1. Automorphisms

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                                                9. Normal Subgroups and Quotient Groups

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                                                11. The Isomorphism Theorems