Group Theory

  1. Cyclic Groups
    1. Definition of a Cyclic Group
      1. Generators of a Group
        1. Definition of a Generator
          1. Finding Generators
            1. Multiple Generators
            2. Properties of Cyclic Groups
              1. Structure of Cyclic Groups
                1. All Subgroups are Cyclic
                  1. Abelian Property
                  2. Fundamental Theorem of Cyclic Groups
                    1. Statement and Proof
                      1. Uniqueness of Subgroups
                      2. Subgroups of Cyclic Groups
                        1. Number and Structure of Subgroups
                          1. Lattice of Subgroups
                            1. Correspondence with Divisors
                            2. Classification of Cyclic Groups
                              1. Infinite Cyclic Groups
                                1. Isomorphism to ℤ
                                2. Finite Cyclic Groups
                                  1. Isomorphism to ℤ/nℤ
                                3. Generators of Finite Cyclic Groups
                                  1. Counting Generators
                                    1. Euler's Totient Function
                                      1. Primitive Roots

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                                    5. Subgroups

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                                    7. Permutation Groups