Abstract Algebra

1. Groups
   1. Definition and Examples
      1. Formal Definition
         1. A group is a set equipped with a binary operation that satisfies certain axioms.
      2. Common Examples
         1. Integers under addition
         2. Non-zero real numbers under multiplication
         3. Symmetric groups modeling permutations
         4. Matrix groups, e.g., GL(n, R), the set of invertible n x n matrices
      3. Non-examples
         1. Non-associative systems
         2. Sets lacking inverses
   2. Group Axioms
      1. Closure
         1. Definition and intuitive explanation
         2. Illustrating with examples, e.g., symmetry operations
      2. Associativity
         1. Explanation and necessity in group theory
         2. Examples of demonstrating associativity in different contexts
      3. Identity Element
         1. Role as a neutral element
         2. Unique existence in a group
         3. Examples in various groups, such as zero in addition
      4. Inverses
         1. Existence of inverse elements
         2. Unique inverses for each element
         3. Examples like additive inverses
   3. Subgroups
      1. Definition and Examples
         1. Closed subset of a group satisfying group axioms
         2. Examples: Even integers within integers
         3. Proper subgroups and trivial subgroups
      2. Criteria for Subgroups
         1. Containing identity element
         2. Closed under group operation
         3. Closed under taking inverses
      3. Normal Subgroups
         1. Definition and motivation
         2. Examples of normal subgroups
         3. Normality Criterion
            1. Using conjugation to test normality
            2. Commutative nature of subgroup elements across group
         4. Quotient Groups
            1. Construction of quotient groups from normal subgroups
            2. Examples and applications of quotient groups in simplifying group analysis
   4. Group Homomorphisms
      1. Definition and Properties
         1. Structure-preserving maps between groups
         2. Examples of homomorphisms, such as determinant functions on matrix groups
      2. Kernel and Image
         1. Kernel as the subset mapping to the identity
         2. Image of the homomorphism and its properties
         3. Relationship between kernel and injectivity
      3. Isomorphisms
         1. Bijective homomorphisms
         2. Criteria for group equivalence, aka isomorphic groups
         3. Real-world examples of isomorphic groups
   5. Symmetry and Group Actions
      1. Permutation Groups
         1. Basics of permutations and cycles
         2. Notations: cycle notation and permutation matrix representation
      2. Symmetric and Alternating Groups
         1. Symmetric group as permutations on n symbols
         2. Alternating group as even permutations
         3. Properties and significance
      3. Group Actions and Orbits
         1. Group actions on sets
         2. Stabilizers and orbit decomposition
         3. Examples: actions on geometric figures or graphs
   6. Special Classes of Groups
      1. Cyclic Groups
         1. General form and generation by a single element
         2. Properties of finite cyclic groups
         3. Application: symmetry of regular polygons
      2. Abelian Groups
         1. Commutative property among elements
         2. Connection to vector spaces and modules
         3. Examples: real numbers under addition
      3. Simple Groups
         1. Nontrivial groups with no normal subgroups apart from identity and self
         2. Classification attempts, examples, and significance
   7. Group Theorems
      1. Lagrange's Theorem
         1. Subgroup order divides group order
         2. Applications in proving existence of elements of a given order
      2. Cauchy's Theorem
         1. Existence of elements of prime order in finite group
         2. Proof sketch and implications
      3. Sylow Theorems
         1. Existence of Sylow p-subgroups
         2. Counting such subgroups and conjugacy
         3. Application in group structure analysis