Group and Simple Examples | BSc Mathematics | Abstract Algebra | Part-2 | Malayalam.

Introduction

This tutorial provides a comprehensive guide to understanding groups and simple examples in the context of abstract algebra, as discussed in the video by ANEES A J. Whether you're a BSc Mathematics student or someone interested in abstract algebra, this step-by-step approach will simplify complex concepts and help you grasp the foundational elements of group theory.

Step 1: Understanding Groups

• Definition of a Group

  • A group is a set G combined with an operation * that satisfies four fundamental properties:
    • Closure: For every a, b in G, the result of the operation a * b is also in G.
    • Associativity: For all a, b, c in G, (a * b) * c = a * (b * c).
    • Identity Element: There exists an element e in G such that for every a in G, e * a = a * e = a.
    • Inverse Element: For every a in G, there exists an element b in G such that a * b = b * a = e.

• Practical Tip

  • When trying to identify a group, always check for these four properties using examples from your studies.

Step 2: Examples of Groups

• Example 1: Integer Addition

  • Set: G = {0, 1, 2, ...} (the set of all integers under addition)
  • Operation: a + b
  • Properties:
    • Closure: The sum of any two integers is an integer.
    • Associativity: (a + b) + c = a + (b + c).
    • Identity: The identity element is 0, since a + 0 = a.
    • Inverse: For any integer a, the inverse is -a, since a + (-a) = 0.

• Example 2: Non-zero Rational Numbers under Multiplication

  • Set: G = {x ∈ Q | x ≠ 0}
  • Operation: a * b
  • Properties:
    • Closure: The product of any two non-zero rational numbers is also a non-zero rational number.
    • Associativity: (a * b) * c = a * (b * c).
    • Identity: The identity is 1, since a * 1 = a.
    • Inverse: The inverse of a is 1/a, since a * (1/a) = 1.

Step 3: Simple Groups

• Definition of Simple Groups

  • A simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself.

• Identifying Simple Groups

  • Check if the group has any nontrivial normal subgroups.
  • Common examples include:
    • The alternating group A5, which is the group of even permutations on five elements.

Step 4: Applications of Group Theory

• Real-World Applications

  • Cryptography: Group theory underpins many encryption algorithms.
  • Chemistry: Symmetry groups help in studying molecular structures.
  • Physics: Group theory is essential in quantum mechanics and particle physics.

• Common Pitfalls to Avoid

  • Confusing normal subgroups with regular subgroups.
  • Forgetting to verify all group properties when working with new examples.

Conclusion

This tutorial has walked you through the fundamental concepts of groups and simple groups in abstract algebra. By understanding the definition and properties of groups, examining practical examples, and recognizing applications of group theory, you should now have a solid foundation to build upon. For further study, explore more complex groups and their properties, or delve into their applications in various fields such as cryptography and physics.