Group Theory

  1. Group Actions
    1. Definition of a Group Action
      1. Left Actions
        1. Right Actions
          1. Faithful Actions
          2. Examples of Group Actions
            1. Left Regular Action
              1. Right Regular Action
                1. Action by Conjugation
                  1. Action on Cosets
                    1. Action on Subgroups
                      1. Linear Actions
                      2. Orbits
                        1. Definition and Properties
                          1. Orbit Decomposition
                            1. Transitive Actions
                            2. Stabilizers
                              1. Definition and Properties
                                1. Isotropy Subgroups
                                  1. Pointwise and Setwise Stabilizers
                                  2. The Orbit-Stabilizer Theorem
                                    1. Statement and Proof
                                      1. Applications and Examples
                                        1. Counting Orbits
                                        2. Fixed Points
                                          1. Definition and Examples
                                            1. Fixed Point Sets
                                              1. Free Actions
                                              2. Burnside's Lemma
                                                1. Statement and Proof
                                                  1. Cauchy-Frobenius Lemma
                                                    1. Applications to Counting
                                                      1. Pólya Enumeration
                                                      2. Conjugacy and Class Equation
                                                        1. Conjugacy Classes
                                                          1. Class Equation
                                                            1. Applications to p-Groups

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                                                          13. The Sylow Theorems