groups and Abelian groups

 In discrete structures, particularly in abstract algebra, a group is a fundamental algebraic structure that consists of a set equipped with an operation that satisfies four key properties:

Definition of a Group

A group (G, ) is a set G with a binary operation **()** that satisfies the following properties:

1. Closure: For all a, b ∈ G, the result of the operation a * b is also in G.

   $$a * b \in G, \quad \forall a, b \in G$$

2. Associativity: For all a, b, c ∈ G, the operation is associative.

   $$(a * b) * c = a * (b * c)$$

3. Identity Element: There exists an element e ∈ G such that for all a ∈ G,

   $$a * e = e * a = a$$

   This element e is called the identity element.

4. Inverse Element: For every a ∈ G, there exists an element a⁻¹ ∈ G such that:

   $$a * a⁻¹ = a⁻¹ * a = e$$

   where a⁻¹ is the inverse of a.

Types of Groups

1. Abelian (Commutative) Group: If a group also satisfies the commutative property,

   $$a * b = b * a, \quad \forall a, b \in G$$

   then it is called an Abelian group (named after Niels Henrik Abel).
   Example: (Z, +), the set of integers under addition.

2. Non-Abelian (Non-Commutative) Group: If a * b ≠ b * a for some elements, the group is non-Abelian.
   Example: Matrix multiplication in GL(n, R) (General Linear Group).

Examples of Groups

1. Integers under Addition: (ℤ, +)

   • Closure: a + b ∈ ℤ
   • Associativity: (a + b) + c = a + (b + c)
   • Identity: 0
   • Inverse: For each a, there exists -a

2. Real Numbers under Multiplication (excluding 0): (ℝ{0}, ×)

   • Closure: a × b ∈ ℝ{0}
   • Associativity: (a × b) × c = a × (b × c)
   • Identity: 1
   • Inverse: Each a has an inverse 1/a

3. Symmetric Group (Sₙ): The set of all permutations of n elements under composition.

Special Types of Groups

• Cyclic Group: Generated by a single element g, i.e., all elements can be written as powers of g.
• Dihedral Group: Represents symmetries of a polygon (rotations and reflections).
• Matrix Groups: Groups like GL(n, R) (General Linear Group) contain invertible n×n matrices under multiplication.