Abstract Algebra

  1. Group Theory
    1. Introduction to Algebraic Structures
      1. Binary Operations
        1. Definition and Examples
          1. Closure Property
          2. Properties of Binary Operations
            1. Associativity
              1. Commutativity
                1. Identity Element
                  1. Inverse Elements
                  2. Semigroups
                    1. Definition of a Semigroup
                      1. Examples and Properties
                      2. Monoids
                        1. Definition of a Monoid
                          1. Examples and Properties
                        2. Groups
                          1. Definition of a Group
                            1. Axioms of a Group
                              1. Closure
                                1. Associativity
                                  1. Identity
                                    1. Inverses
                                    2. Basic Properties and Theorems
                                      1. Uniqueness of Identity
                                        1. Uniqueness of Inverses
                                          1. Cancellation Laws
                                          2. Order of a Group
                                            1. Definition and Properties
                                            2. Order of an Element
                                              1. Definition and Properties
                                                1. Relationship to Group Order
                                                2. Abelian Groups
                                                  1. Definition of Abelian Group
                                                    1. Examples and Properties
                                                    2. Non-Abelian Groups
                                                      1. Definition and Examples
                                                    3. Examples of Groups
                                                      1. Number Systems under Addition
                                                        1. Integers
                                                          1. Rational Numbers
                                                            1. Real Numbers
                                                              1. Complex Numbers
                                                              2. Number Systems under Multiplication
                                                                1. Nonzero Rational Numbers
                                                                  1. Nonzero Real Numbers
                                                                    1. Nonzero Complex Numbers
                                                                    2. The Integers Modulo n
                                                                      1. Additive Group Z_n
                                                                        1. Multiplicative Group U(n)
                                                                        2. Groups of Matrices
                                                                          1. General Linear Group GL(n, R)
                                                                            1. Special Linear Group SL(n, R)
                                                                              1. Orthogonal Groups
                                                                              2. Groups of Permutations
                                                                                1. Symmetric Groups S_n
                                                                                  1. Properties and Structure
                                                                                  2. Groups of Symmetries
                                                                                    1. Dihedral Groups D_n
                                                                                      1. Symmetries of Regular Polygons
                                                                                      2. The Klein Four-Group
                                                                                        1. Structure and Properties
                                                                                          1. Isomorphism with Z_2 × Z_2
                                                                                        2. Subgroups
                                                                                          1. Definition of a Subgroup
                                                                                            1. Subgroup Test
                                                                                              1. One-Step Subgroup Test
                                                                                                1. Two-Step Subgroup Test
                                                                                                2. Center of a Group
                                                                                                  1. Definition and Properties
                                                                                                  2. Centralizer of an Element
                                                                                                    1. Definition and Properties
                                                                                                    2. Cyclic Subgroups
                                                                                                      1. Generation by a Single Element
                                                                                                        1. Properties of Cyclic Subgroups
                                                                                                      2. Cyclic Groups
                                                                                                        1. Definition and Properties
                                                                                                          1. Generators of a Cyclic Group
                                                                                                            1. Fundamental Theorem of Cyclic Groups
                                                                                                              1. Classification of Subgroups of a Cyclic Group
                                                                                                                1. Structure of Finite Cyclic Groups
                                                                                                                  1. Structure of Infinite Cyclic Groups
                                                                                                                  2. Permutation Groups
                                                                                                                    1. Symmetric Group S_n
                                                                                                                      1. Definition and Structure
                                                                                                                        1. Order of S_n
                                                                                                                        2. Cycle Notation
                                                                                                                          1. Writing Permutations as Cycles
                                                                                                                            1. Disjoint Cycles
                                                                                                                              1. Order of a Permutation
                                                                                                                              2. Transpositions
                                                                                                                                1. Definition and Properties
                                                                                                                                  1. Expressing Permutations as Products of Transpositions
                                                                                                                                  2. The Alternating Group A_n
                                                                                                                                    1. Definition and Properties
                                                                                                                                      1. Even and Odd Permutations
                                                                                                                                      2. Cayley's Theorem
                                                                                                                                        1. Statement and Proof
                                                                                                                                      3. Cosets and Lagrange's Theorem
                                                                                                                                        1. Left Cosets
                                                                                                                                          1. Definition and Examples
                                                                                                                                          2. Right Cosets
                                                                                                                                            1. Definition and Examples
                                                                                                                                            2. Properties of Cosets
                                                                                                                                              1. Partitioning the Group
                                                                                                                                                1. Equivalence Relations
                                                                                                                                                2. Index of a Subgroup
                                                                                                                                                  1. Definition and Calculation
                                                                                                                                                  2. Lagrange's Theorem
                                                                                                                                                    1. Statement and Proof
                                                                                                                                                      1. Consequences for Element Orders
                                                                                                                                                    2. Homomorphisms and Isomorphisms
                                                                                                                                                      1. Group Homomorphisms
                                                                                                                                                        1. Definition and Examples
                                                                                                                                                          1. Properties of Homomorphisms
                                                                                                                                                          2. Kernel of a Homomorphism
                                                                                                                                                            1. Definition and Properties
                                                                                                                                                            2. Image of a Homomorphism
                                                                                                                                                              1. Definition and Properties
                                                                                                                                                              2. Group Isomorphisms
                                                                                                                                                                1. Definition and Examples
                                                                                                                                                                  1. Properties of Isomorphic Groups
                                                                                                                                                                  2. Automorphisms
                                                                                                                                                                    1. Definition of Automorphism
                                                                                                                                                                      1. Automorphism Group
                                                                                                                                                                      2. Inner Automorphisms
                                                                                                                                                                        1. Definition and Properties
                                                                                                                                                                      3. Normal Subgroups and Quotient Groups
                                                                                                                                                                        1. Definition of a Normal Subgroup
                                                                                                                                                                          1. Tests for Normality
                                                                                                                                                                            1. Conjugation Criterion
                                                                                                                                                                              1. Coset Criterion
                                                                                                                                                                              2. Quotient Groups
                                                                                                                                                                                1. Construction and Properties
                                                                                                                                                                                  1. Well-Definedness
                                                                                                                                                                                  2. Properties of Quotient Groups
                                                                                                                                                                                    1. Order of Quotient Groups
                                                                                                                                                                                  3. The Isomorphism Theorems
                                                                                                                                                                                    1. The First Isomorphism Theorem
                                                                                                                                                                                      1. Statement and Proof
                                                                                                                                                                                      2. The Second Isomorphism Theorem
                                                                                                                                                                                        1. Statement and Proof
                                                                                                                                                                                        2. The Third Isomorphism Theorem
                                                                                                                                                                                          1. Statement and Proof
                                                                                                                                                                                          2. The Lattice Isomorphism Theorem
                                                                                                                                                                                            1. Statement and Applications
                                                                                                                                                                                          3. Group Actions
                                                                                                                                                                                            1. Definition of a Group Action on a Set
                                                                                                                                                                                              1. Examples of Group Actions
                                                                                                                                                                                                1. Orbits
                                                                                                                                                                                                  1. Definition and Properties
                                                                                                                                                                                                  2. Stabilizers
                                                                                                                                                                                                    1. Definition and Properties
                                                                                                                                                                                                    2. The Orbit-Stabilizer Theorem
                                                                                                                                                                                                      1. Statement and Applications
                                                                                                                                                                                                      2. Burnside's Lemma
                                                                                                                                                                                                        1. Statement and Proof
                                                                                                                                                                                                          1. Counting Orbits
                                                                                                                                                                                                        2. The Sylow Theorems
                                                                                                                                                                                                          1. p-Groups
                                                                                                                                                                                                            1. Definition and Properties
                                                                                                                                                                                                            2. Sylow p-Subgroups
                                                                                                                                                                                                              1. Definition and Existence
                                                                                                                                                                                                              2. The First Sylow Theorem
                                                                                                                                                                                                                1. Statement and Proof
                                                                                                                                                                                                                2. The Second Sylow Theorem
                                                                                                                                                                                                                  1. Statement and Proof
                                                                                                                                                                                                                  2. The Third Sylow Theorem
                                                                                                                                                                                                                    1. Statement and Proof
                                                                                                                                                                                                                    2. Applications in Classifying Finite Groups
                                                                                                                                                                                                                    3. Structure of Finite Groups
                                                                                                                                                                                                                      1. Direct Products of Groups
                                                                                                                                                                                                                        1. External Direct Products
                                                                                                                                                                                                                          1. Internal Direct Products
                                                                                                                                                                                                                            1. Properties and Examples
                                                                                                                                                                                                                            2. The Fundamental Theorem of Finitely Generated Abelian Groups
                                                                                                                                                                                                                              1. Statement and Structure
                                                                                                                                                                                                                                1. Primary Decomposition
                                                                                                                                                                                                                                  1. Invariant Factor Form
                                                                                                                                                                                                                                  2. Solvable Groups
                                                                                                                                                                                                                                    1. Definition and Examples
                                                                                                                                                                                                                                      1. Composition Series
                                                                                                                                                                                                                                      2. Nilpotent Groups
                                                                                                                                                                                                                                        1. Definition and Properties
                                                                                                                                                                                                                                          1. Central Series
                                                                                                                                                                                                                                          2. Simple Groups
                                                                                                                                                                                                                                            1. Definition and Examples
                                                                                                                                                                                                                                              1. Classification Overview

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                                                                                                                                                                                                                                          3. Ring Theory