Unit 4: Algebraic Structures

Algebraic Structures 

An algebraic structure is a non-empty set combined with one or more operations that follow certain rules (properties).

In discrete mathematics, algebraic structures help in understanding abstract systems, which are widely used in computer science, cryptography, automata theory, data structures, and algorithms.

Basic Components of Algebraic Structures

An algebraic structure consists of:

1. A non-empty set (denoted by A)
2. One or more binary operations (denoted by ∘, +, *)

Binary Operation: An operation ∘ on set A is called binary if:

A × A → A

Properties of Algebraic Structures

1. Closure Property

For all a, b ∈ A, a ∘ b ∈ A

2. Associative Property

(a ∘ b) ∘ c = a ∘ (b ∘ c)

3. Commutative Property

a ∘ b = b ∘ a

4. Identity Element

An element e ∈ A such that:

a ∘ e = e ∘ a = a

5. Inverse Element

For each a ∈ A, there exists b ∈ A such that:

a ∘ b = b ∘ a = e

Types of Algebraic Structures

Semigroup

A semigroup is an algebraic structure (A, ∘) where:

1. A is a non-empty set
2. ∘ is a binary operation on A
3. ∘ is associative

Example

• (N, +) is a semigroup
• (Z, −) is NOT a semigroup (not associative)

Monoid

A monoid is a semigroup with an identity element.

Conditions

1. Closure
2. Associativity
3. Identity element exists

Example

• (N, +) with identity 0
• (N, ×) with identity 1

Group

A group is a monoid in which every element has an inverse.

Conditions

1. Closure
2. Associativity
3. Identity element
4. Inverse element

Example of Group

• (Z, +)
• (Q, +)

Non-Example

• (N, +) is not a group (no inverse)

Abelian Group

A group is called an Abelian group if the operation is commutative.

Condition

a ∘ b = b ∘ a

Example

• (Z, +)
• (Q, +)

Comparison of Algebraic Structures

Property	Semigroup	Monoid	Group	Abelian Group
Closure	Yes	Yes	Yes	Yes
Associative	Yes	Yes	Yes	Yes
Identity	No	Yes	Yes	Yes
Inverse	No	No	Yes	Yes
Commutative	May or may not	May or not	May or not	Yes

Properties of Group (Very Important)

1. Uniqueness of Identity

A group has only one identity element.

2. Uniqueness of Inverse

Each element in a group has exactly one inverse.

3. Cancellation Law

If a ∘ b = a ∘ c, then b = c

4. Inverse of Inverse

(a⁻¹)⁻¹ = a

5. Identity is Self-Inverse

e⁻¹ = e

6. Inverse of a Product

(a ∘ b)⁻¹ = b⁻¹ ∘ a⁻¹

7. Subgroup (Short Introduction)

A subset H of group G is called a subgroup if H itself forms a group under the same operation.

Exam-Oriented Important Points

• Definitions are frequently asked
• Properties of groups carry high marks
• Examples and non-examples are important
• Comparison tables help in short answers

Common Exam Questions

1. Define semigroup with example
2. Differentiate between monoid and group
3. What is an Abelian group?
4. Prove uniqueness of identity in a group
5. Give examples of algebraic structures

Conclusion

Algebraic structures form the foundation of abstract algebra and are essential for understanding advanced topics in computer science. A clear understanding of semigroup, monoid, group, and Abelian group is crucial for MCA students.

Subgroup

Let (G, ∘) be a group. A subset H ⊆ G is called a subgroup of G if H itself forms a group under the operation ∘.

Notation: H ≤ G

Subgroup Tests (Very Important for Exams)

1. Subgroup Criterion (One-Step Test)

A non-empty subset H of G is a subgroup if:

• For all a, b ∈ H, a ∘ b⁻¹ ∈ H

2. Two-Step Test

H is a subgroup if:

1. H is closed under operation
2. For every a ∈ H, a⁻¹ ∈ H

Examples

• (Z, +) has subgroup (2Z, +)
• ({0}, +) is a trivial subgroup

Types of Subgroups

Type	Description
Trivial Subgroup	{e}
Improper Subgroup	G itself
Proper Subgroup	Any subgroup except {e} and G

Cyclic Group

A group G is called cyclic if there exists an element a ∈ G such that:

G = {aⁿ | n ∈ Z}

The element a is called the generator of the group.

Notation

G = ⟨a⟩

Examples

• (Z, +) is a cyclic group generated by 1 or −1
• ({0,1,2,3}, + mod 4) is cyclic

Properties of Cyclic Groups

1. Every cyclic group is Abelian
2. Every subgroup of a cyclic group is cyclic
3. Finite cyclic groups have generators φ(n)

Cosets

Let H be a subgroup of G and a ∈ G.

• Left Coset: aH = {a ∘ h | h ∈ H}
• Right Coset: Ha = {h ∘ a | h ∈ H}

Example

Let G = Z, H = 2Z

Left cosets:

• 0 + 2Z = even integers
• 1 + 2Z = odd integers

Properties of Cosets

• Every element of G belongs to exactly one coset
• All cosets have same number of elements
• Either two cosets are identical or disjoint

Lagrange’s Theorem (Very Important)

If G is a finite group and H is a subgroup of G, then:

|H| divides |G|

Permutation Groups

A permutation is a bijection from a set to itself.

The set of all permutations of n elements forms a group called the symmetric group, denoted by Sₙ.

Example: S₃ = {all permutations of {1,2,3}}

Properties

1. Group operation is composition
2. Identity is the identity permutation
3. Inverse exists for every permutation

Even and Odd Permutations

Type	Description
Even	Can be written as even number of transpositions
Odd	Can be written as odd number of transpositions

Homomorphism of Groups

Let (G, ∘) and (G′, *) be two groups.

A function f: G → G′ is called a homomorphism if:

f(a ∘ b) = f(a) * f(b)

for all a, b ∈ G.

Example: f: (Z, +) → (Z, +) defined by f(x) = 2x is a homomorphism.

Properties of Homomorphism

1. f(e) = e′
2. f(a⁻¹) = [f(a)]⁻¹
3. Kernel of f is a subgroup

Isomorphism of Groups

A homomorphism f: G → G′ is called an isomorphism if it is:

1. One-to-one (injective)
2. Onto (surjective)

If such a mapping exists, G and G′ are said to be isomorphic, written as:

G ≅ G′

Significance: Isomorphic groups have the same algebraic structure.

Example: (Z, +) ≅ (2Z, +)

Difference: Homomorphism vs Isomorphism

Basis	Homomorphism	Isomorphism
Structure preserving	Yes	Yes
One-to-one	Not necessary	Yes
Onto	Not necessary	Yes

Exam-Oriented Key Points

• Subgroup tests are highly important
• Cyclic groups and generators are frequently asked
• Lagrange’s theorem is compulsory
• Definitions of homomorphism and isomorphism carry direct marks

Common MCA Exam Questions

1. Define subgroup and state subgroup test
2. Prove that every cyclic group is Abelian
3. Explain cosets with example
4. Define permutation group
5. Differentiate homomorphism and isomorphism

Conclusion

Subgroups, cyclic groups, cosets, and group mappings are core concepts in group theory. These topics are highly scoring and conceptually important for MCA students and competitive exams.

Rings and Fields

Rings and Fields are important algebraic structures that extend the concept of groups by introducing two binary operations. These structures are widely used in computer science, cryptography, coding theory, databases, and numerical computation.

For MCA students, questions from this topic are generally definition-based, property-based, and comparison-based, making it a high-scoring unit.

Ring

A ring is an algebraic structure (R, +, ·) consisting of a non-empty set R with two binary operations addition (+) and multiplication (·) such that:

1. (R, +) is an Abelian group
2. (R, ·) is closed and associative
3. Multiplication is distributive over addition

Mathematically:

• a · (b + c) = (a · b) + (a · c)
• (a + b) · c = (a · c) + (b · c)

Properties of Ring (Elementary Properties)

For all a, b, c ∈ R:

1. a + 0 = a (Additive identity)
2. a + (−a) = 0 (Additive inverse)
3. a · 0 = 0 · a = 0
4. (−a) · b = a · (−b) = −(a · b)

Types of Rings

1. Commutative Ring

A ring R is called commutative if:

a · b = b · a

Example: (Z, +, ·)

2. Ring with Identity

A ring has an identity element (1) if:

a · 1 = 1 · a = a

Example: (Z, +, ·)

3. Integral Domain

A commutative ring with identity and no zero divisors.

Zero divisor: a · b = 0 where a ≠ 0 and b ≠ 0

Example: (Z, +, ·)

Field

A field is an algebraic structure (F, +, ·) such that:

1. (F, +) is an Abelian group
2. (F − {0}, ·) is an Abelian group
3. Multiplication is distributive over addition

Elementary Properties of Field

For all a, b ∈ F, a ≠ 0:

1. a + 0 = a
2. a · 1 = a
3. a + (−a) = 0
4. a · a⁻¹ = 1
5. a · 0 = 0

Examples of Fields

• Rational numbers (Q)
• Real numbers (R)
• Complex numbers (C)
• Finite field Zₚ (p is prime)

Non-Example

• Integers (Z) is not a field (no multiplicative inverse for all elements)

Difference Between Ring and Field (Very Important)

Basis	Ring	Field
Operations	Two (+, ·)	Two (+, ·)
Additive group	Abelian	Abelian
Multiplicative inverse	Not required	Required for all non-zero
Commutative multiplication	Not compulsory	Compulsory

Relationship Among Algebraic Structures

Group ⊂ Ring ⊂ Field

(Every field is a ring, every ring is based on a group)

Important Exam Points

• Definition of ring and field are frequently asked
• Properties should be written point-wise
• Comparison table is compulsory for full marks
• Examples and non-examples improve answers

Common MCA Exam Questions

1. Define ring with example
2. Define field and give examples
3. Differentiate between ring and field
4. What is an integral domain?
5. Explain elementary properties of a field

Conclusion

Rings and fields form the backbone of algebraic theory used in computer science. A strong understanding of definitions and properties is essential for MCA students to score well in examinations.