Group Theory

  1. Normal Subgroups and Quotient Groups
    1. Definition of a Normal Subgroup
      1. Normal Subgroup Criterion
        1. Conjugation Characterization
          1. Examples of Normal Subgroups
          2. Normality Tests
            1. Conjugation Test
              1. Index 2 Subgroups
                1. Kernel Test
                  1. Center and Derived Subgroup
                  2. Conjugation
                    1. Conjugate Elements
                      1. Conjugate Subgroups
                        1. Conjugacy Classes
                        2. The Quotient Group Construction
                          1. Definition of Coset Multiplication
                            1. Well-definedness of the Operation
                              1. Verification of Group Axioms
                              2. Examples of Quotient Groups
                                1. ℤ/nℤ as a Quotient Group
                                  1. Dₙ/⟨r⟩
                                    1. Sₙ/Aₙ ≅ ℤ/2ℤ
                                    2. Properties of Quotient Groups
                                      1. Order of Quotient Groups
                                        1. Abelian Quotients
                                        2. The Simple Group Concept
                                          1. Definition of a Simple Group
                                            1. Examples of Simple Groups
                                              1. Simplicity of Aₙ for n ≥ 5
                                                1. Classification Overview

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                                              8. Cosets and Lagrange's Theorem

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                                              10. Group Homomorphisms