Unit 2947: Abstract Algebra and Its Applications

1. Introduction to Groups

• Definition of a group
• Group axioms:
  • Closure
  • Associativity
  • Identity element
  • Inverses
• Examples of groups (e.g., integers under addition, permutation groups)
• The group operation and notation

2. Subgroups and Lagrange's Theorem

• Definition of a subgroup
• Criteria for subgroups
• Examples of subgroups
• Statement and proof of Lagrange's Theorem
• Consequences of Lagrange's Theorem (order of elements divides order of group)
• Cosets and index of a subgroup

3. Group Homomorphisms and Isomorphisms

• Definition of group homomorphisms
• Kernel and image of a homomorphism
• Properties of homomorphisms
• Definition of isomorphisms
• Criteria for isomorphisms (bijectivity)
• Examples illustrating homomorphisms and isomorphisms

4. Rings and Fields

• Definition and examples of rings
• Ring properties (associativity, distributivity, additive identity, etc.)
• Ring homomorphisms
• Ideals and their significance
• Integral domains
• Definition and properties of fields
• Examples of fields (rational numbers, real numbers, finite fields)

5. Polynomial Rings

• Construction of polynomial rings over a ring
• Polynomial addition, multiplication, and degree
• Irreducible polynomials
• Factorization of polynomials
• Polynomial division algorithm
• Applications and examples

6. Vector Spaces

• Definition and axioms of vector spaces over a field
• Examples of vector spaces
• Subspaces
• Linear combinations, span, and linear independence
• Bases and dimension

7. Linear Transformations and Matrices

• Definition of linear transformations between vector spaces
• Kernel and image of a linear transformation
• Rank and nullity theorem
• Matrix representation of linear transformations
• Matrix operations (addition, multiplication, inverse)

8. Group Actions

• Definition of group actions on sets
• Orbits and stabilizers
• Examples of group actions
• Applications of group actions (e.g., symmetry groups, counting arguments)

9. Applications of Abstract Algebra

• Cryptography:
  • Use of groups and fields in encryption algorithms
  • Public-key cryptography basics
• Coding Theory:
  • Error detection and correction codes
  • Role of algebraic structures
• Computer Science:
  • Automata theory and formal languages
  • Algebraic data structures
• Physics:
  • Symmetries and conservation laws
  • Group theory in quantum mechanics