Group Theory

Group theory is a branch of abstract algebra that studies algebraic structures called groups, which are sets equipped with a single operation satisfying a few fundamental axioms like closure, associativity, the existence of an identity element, and the existence of an inverse for every element. At its core, group theory is the mathematical study of symmetry, providing a formal framework to analyze transformations that leave an object or system invariant. This powerful concept is foundational to numerous areas of science and mathematics, enabling the classification of crystal structures in chemistry, the description of fundamental particles in physics, and the development of secure algorithms in cryptography.

1. Introduction to Algebraic Structures
   1. Sets and Operations
      1. Definition of a Set
      2. Set Notation and Terminology
      3. Cartesian Product of Sets
      4. Functions and Mappings between Sets
      5. Equivalence Relations
      6. Partitions of Sets
   2. Binary Operations
      1. Definition of a Binary Operation
      2. Examples of Binary Operations
      3. Tables for Finite Binary Operations
      4. Closure under Operations
   3. Properties of Binary Operations
      1. Associativity
         1. Definition and Examples
         2. Non-associative Operations
         3. Verification Methods
      2. Commutativity
         1. Definition and Examples
         2. Non-commutative Operations
      3. Identity Elements
         1. Definition of Identity
         2. Left and Right Identities
         3. Existence and Uniqueness
      4. Inverse Elements
         1. Definition of Inverse
         2. Left and Right Inverses
         3. Existence and Uniqueness
   4. Semigroups and Monoids
      1. Definition of a Semigroup
      2. Examples of Semigroups
      3. Definition of a Monoid
      4. Examples of Monoids
      5. Relationship between Semigroups, Monoids, and Groups