Group Theory

  1. Definition of a Group
    1. The Group Axioms
      1. Closure
        1. Definition and Verification
          1. Checking Closure in Examples
          2. Associativity
            1. Application in Groups
              1. Verification Techniques
              2. Existence of an Identity Element
                1. Uniqueness of Identity
                  1. Proof of Uniqueness
                  2. Existence of an Inverse Element
                    1. Uniqueness of Inverses
                      1. Proof of Uniqueness
                    2. Basic Properties and Theorems
                      1. Cancellation Laws
                        1. Left Cancellation Law
                          1. Right Cancellation Law
                            1. Proofs and Applications
                            2. Inverse Properties
                              1. Inverse of an Inverse
                                1. Inverse of Products
                                2. Powers of Elements
                                  1. Definition of Powers
                                    1. Laws of Exponents
                                  2. Abelian vs. Non-Abelian Groups
                                    1. Definition of Abelian Group
                                      1. Examples of Abelian Groups
                                        1. Examples of Non-Abelian Groups
                                          1. Commutativity Tests

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                                        3. Fundamental Examples of Groups