Group Theory

  1. Permutation Groups
    1. Definition of a Permutation
      1. Permutations as Bijections
        1. Composition of Permutations
        2. The Symmetric Group
          1. Definition of Sₙ
            1. Structure and Properties
              1. Order of Sₙ
              2. Cycle Notation
                1. Writing Permutations in Cycle Notation
                  1. Disjoint Cycles
                    1. Definition and Properties
                      1. Commutativity of Disjoint Cycles
                      2. Products of Cycles
                        1. Multiplication of Cycles
                          1. Converting between Notations
                        2. Cycle Structure
                          1. Cycle Type
                            1. Order of a Permutation
                              1. Calculation from Cycle Decomposition
                                1. Least Common Multiple Method
                              2. Transpositions
                                1. Definition of a Transposition
                                  1. Every Permutation as a Product of Transpositions
                                    1. Minimal Transposition Decompositions
                                    2. Even and Odd Permutations
                                      1. Definition of Parity
                                        1. Sign of a Permutation
                                          1. Properties of Parity
                                          2. The Alternating Group
                                            1. Definition of Aₙ
                                              1. Properties and Structure
                                                1. Order of Aₙ
                                                  1. Generators of Aₙ
                                                  2. Cayley's Theorem
                                                    1. Statement and Proof
                                                      1. Embedding Groups into Symmetric Groups
                                                        1. Regular Representation

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                                                      6. Cyclic Groups

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                                                      8. Cosets and Lagrange's Theorem