Group Theory

  1. The Sylow Theorems
    1. p-Groups
      1. Definition and Examples
        1. Properties of p-Groups
          1. Centers of p-Groups
            1. Maximal p-Subgroups
            2. Sylow p-Subgroups
              1. Definition and Examples
                1. Maximal p-Subgroups
                2. The First Sylow Theorem
                  1. Statement and Proof
                    1. Existence of Sylow p-Subgroups
                      1. Cauchy's Theorem
                      2. The Second Sylow Theorem
                        1. Statement and Proof
                          1. Conjugacy of Sylow p-Subgroups
                            1. Normalizer Conditions
                            2. The Third Sylow Theorem
                              1. Statement and Proof
                                1. Number of Sylow p-Subgroups
                                  1. Congruence Conditions
                                  2. Applications of the Sylow Theorems
                                    1. Proving Non-simplicity
                                      1. Classifying Groups of Small Order
                                        1. Structure Theorems
                                          1. Counting Arguments

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                                        12. Group Actions

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                                        14. Direct Products and Sums