Abelian Group: Properties, Example, Solved Problems

An Abelian group, named after the mathematician Niels Henrik Abel, is a fundamental concept in abstract algebra. It is a group in which the group operation is commutative, meaning the order of operation does not affect the result. Abelian groups appear in various branches of mathematics and have significant applications in engineering, physics, and computer science.

An Abelian group is a set G equipped with an operation * that combines any two elements a and b to form another element in G, denoted as a * b.

The group must satisfy the following axioms:

1. Closure: For every a, b ∈ G, a ∗ b ∈ G.
2. Associativity: For every a, b, c ∈ G, (a ∗ b) ∗ c = a ∗ (b ∗ c).
3. Identity Element: There exists an element e ∈ G such that for every a ∈ G, a ∗ e = e ∗ a =a.
4. Inverse Element: For each a ∈ G, there exists an element b ∈ G such that a ∗ b = b ∗ a = e.
5. Commutativity: For every a, b ∈ G, a ∗ b = b ∗ a.

Examples of Abelian Groups

Integers under Addition

• The set of integers Z with the operation of addition + forms an Abelian group.
• The identity element is 0,
• The inverse of any integer a is −a.

Real Numbers under Addition

• The set of real numbers R with the operation of addition + also forms an Abelian group. 
• The identity element is 0,
• The inverse of any real number a is −a.

Non-Examples

Non-zero Real Numbers under Multiplication

• The set of non-zero real numbers under multiplication ⋅ is not an Abelian group, since the inverse of 0 does not exist.

Properties of Abelian Groups

These are the five properties of Abelian Groups:

Closure Property:
For every pair of elements a, b ∈ G, the result of the operation a∗b is also an element of G: a ∗ b ∈ G.

Associativity:
The group operation * is associative, meaning that for any three elements a, b, c ∈ G: (a ∗ b) ∗ c = a ∗ (b ∗ c).

Identity Element:
There exists an identity element in e∈G such that for every element a ∈ G: a ∗ e = e ∗ a = a.

Inverse Element:
For each element in a∈G, there exists an inverse element b ∈ G such that: a ∗ b = b ∗ a = e.

Commutativity:
An Abelian group is characterized by its commutative property, which means that for any two elements a and b in the group G, the operation * satisfies: a ∗ b = b ∗ a.

Example Problems on Abelian Groups

Prove that (I, +) is an abelian group. i.e. The set of all integers I form an abelian group with respect to binary operation '+'.

Solution-:

Set = I = { .................-3, -2 , -1 , 0, 1, 2 , 3.................. }. Binary Operation= '+' Algebraic Structure= (I, +)

We have to prove that (I, +) is an abelian group. To prove that set of integers I is an abelian group we must satisfy the following five properties that is:

• Closure Property,
• Associative Property,
• Identity Property,
• Inverse Property, and
• Commutative Property.

1) Closure Property

∀ a , b ∈ I ⇒ a + b ∈ I 2, -3 ∈ I ⇒ -1 ∈ I

Hence Closure Property is satisfied.

2) Associative Property

( a + b ) + c = a + ( b + c) ∀ a , b , c ∈ I 2 ∈ I, -6 ∈ I , 8 ∈ I So, LHS = ( a + b ) +c = (2 + ( -6 ) ) + 8 = 4 RHS= a + ( b + c ) =2 + ( - 6 + 8 ) = 4
Hence RHS = LHS

Associative Property is also Satisfied.

3) Identity Property

a + 0 = a ∀ a ∈ I , 0 ∈ I 5 ∈ I 5+0 = 5 -17 ∈ I -17 + 0 = - 17

Identity property is also satisfied.

4) Inverse Property

a + ( -a ) = 0 ∀ a ∈ I , -a ∈ I ,0 ∈ I a=18 ∈ I then ∋ a number -18 such that 18 + ( -18 ) = 0

So, Inverse property is also satisfied.

5) Commutative Property

a + b = b + a ∀ a , b ∈ I Let a = 19, b = 20 LHS = a + b = 19 + ( -20 ) = -1 RHS = b + a = -20 + 19 = -1 LHS = RHS

Commutative Property is also satisfied. We can see that all five property is satisfied. Hence (I, +) is an Abelian Group.

Note-: (I, +) is also Groupoid, Monoid, and Semigroup.

Applications in Engineering

Signal Processing

In signal processing, the Fourier transform, which decomposes a signal into its constituent frequencies, relies on the properties of Abelian groups. The set of complex numbers under addition forms an Abelian group, facilitating the mathematical operations involved.

Cryptography

Abelian groups play a crucial role in cryptography, particularly in the construction of elliptic curve cryptography (ECC). The points on an elliptic curve form an Abelian group, and their properties are used to create secure cryptographic algorithms.

Physics

In physics, the symmetry operations of a crystal lattice form an Abelian group. This helps in understanding the physical properties of materials and in the development of theories like group theory in quantum mechanics.