Group Theory

Back to Discrete Mathematics
Arjit Sharma
01 Min

A branch of mathematics that studies elements present in a group. A group is a algebraic structure, and a algebraic structure is a set of elements with a particular operation and some rules. I know you didnt quite get it, so lets dive deeper.

Important Properties

  • Closure Property - For all a, b in set S, a * b is in S.
  • Associative Property - (a * b) * c = a * (b * c)
  • Identity Property - There exists an element e where a * e = e * a = a
  • Inverse Property - For each element a, there exists -a where a * (-a) = (-a) * a = e
  • Commutative Property - a * b = b * a.

Note that here * means any operator !!

What are Algebraic Structures ?

Algebraic structures defines a set of values and rules for combining them. There are many types, some common ones are -

1. Semigroup

If a set with respect to a binary operation * satisfies Closure and Associative property.

example → (E,x) : Multiplication on Even numbers.

2. Monoid

Properties of a Semigroup + Identity property.

example → (N,x) : Multiplication on Natural numbers.

3. Group

Properties of a Monoid + Inverse property.

example → (Z,+) : Addition on Integral numbers.

4. Abelian Group

Properties of a Group + Commutative property.

example → (Z,+): Addition on Integers.

Practical Application

Algebraic structures are a theoretical concept but they provide practical pattern for solving real-world problems. The properties being followed by a algebraic structure allows use to shape programming model.

For example → Concatenation of Strings often follow properties of a Monoid

// Closure
"hello" + "world" = "helloworld" // Always results in a String

// Associative
("hello" + "world") + "!" = "hello" + ("world" + "!")

// Identity
"" + "hello" = "hello"

Conclusion

Group theory provides a robust mathematical framework for understanding and organizing computational structures. The properties (closure, associativity, identity, inverse) serve as guarantees for system behavior and are seen all around us.

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