Group Theory

  1. Cosets and Lagrange's Theorem
    1. Left and Right Cosets
      1. Definition of Left Coset
        1. Definition of Right Coset
          1. Coset Representatives
          2. Properties of Cosets
            1. Partitioning the Group
              1. Equal Size of Cosets
                1. Coset Equality Conditions
                2. Index of a Subgroup
                  1. Definition and Calculation
                    1. Finite and Infinite Index
                    2. Lagrange's Theorem
                      1. Statement and Proof
                        1. Geometric Interpretation
                        2. Corollaries of Lagrange's Theorem
                          1. Order of an Element Divides Order of the Group
                            1. Groups of Prime Order are Cyclic
                              1. Subgroups of Prime Index
                                1. Fermat's Little Theorem
                                  1. Euler's Theorem
                                  2. Applications of Lagrange's Theorem
                                    1. Counting Arguments
                                      1. Non-existence Results

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                                    7. Permutation Groups

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                                    9. Normal Subgroups and Quotient Groups