Discrete Structure Unit 2
Algebraic Structures

1. Algebraic Structure

An algebraic structure is a non-empty set together with one or more operations defined on it.

Example: (Z, +) means integers with addition operation.

2. Binary Operation

A binary operation combines two elements of a set and gives another element of the same set.

Here, addition is a binary operation on integers because result is also an integer.

3. Properties of Algebraic Structures

• Closure: If a, b ∈ A, then a * b ∈ A.
• Associative: (a * b) * c = a * (b * c).
• Identity: There exists e such that a * e = e * a = a.
• Inverse: For every a, there exists a⁻¹ such that a * a⁻¹ = e.
• Commutative: a * b = b * a.

4. Semigroup

A non-empty set S with binary operation * is called a semigroup if:

• Closure property holds.
• Associative property holds.

5. Monoid

A semigroup with identity element is called a monoid.

• Closure property
• Associative property
• Identity element exists

6. Group

A non-empty set G with binary operation * is called a group if it satisfies:

• Closure property
• Associative property
• Identity element exists
• Inverse element exists

7. Abelian Group

A group is called an abelian group if commutative property holds.

Example: (Z, +) is abelian because a + b = b + a.

Every abelian group is a group, but every group is not necessarily abelian.

8. Subgroup

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

Even integers form a subgroup of integers under addition.

9. Cyclic Group

A group generated by a single element is called a cyclic group.

• Every cyclic group is abelian.
• A cyclic group can be finite or infinite.
• Example: (Z, +) is generated by 1.

10. Cosets

Let H be a subgroup of group G and a ∈ G.

• Left Coset: aH = { ah : h ∈ H }
• Right Coset: Ha = { ha : h ∈ H }

Cosets are important in group theory and normal subgroup questions.

11. Normal Subgroup

A subgroup H of G is called a normal subgroup if left coset and right coset are equal for every element of G.

12. Homomorphism

A homomorphism is a structure-preserving mapping between two algebraic structures.

It preserves the operation of the group.

13. Isomorphism

An isomorphism is a bijective homomorphism. It shows that two algebraic structures have the same structure.

• It must be one-one.
• It must be onto.
• It must preserve operation.

14. Ring

A ring is an algebraic structure with two binary operations, generally addition and multiplication.

• Addition forms an abelian group.
• Multiplication is associative.
• Distributive laws hold.

15. Field

A field is a ring in which every non-zero element has multiplicative inverse.

• Addition and multiplication are defined.
• Every non-zero element has inverse under multiplication.
• Distributive laws hold.

1. Define group and abelian group with examples.
2. Define subgroup. Prove that even integers form subgroup of integers.
3. Define cyclic group with example.
4. Define cosets with suitable example.
5. Explain normal subgroup with example.
6. Define homomorphism and isomorphism with examples.
7. Explain rings and fields with examples.
8. Explain semigroup and monoid with examples.
9. Explain properties of groups.
10. Explain permutation group with example.

Most Important Topics

Most Expected Questions

• Define group and abelian group with examples.
• Prove that even integers form subgroup of integers.
• Explain cyclic group and prove every cyclic group is abelian.
• Explain cosets with example.
• Define homomorphism and isomorphism.
• Explain rings and fields with examples.
• Explain normal subgroup.

• First prepare Group and Abelian Group because it is highly important.
• Do not skip Subgroup, especially even integers subgroup proof.
• Prepare Homomorphism and Isomorphism for theory questions.
• Rings and Fields are very important for 14-mark answers.
• Prepare Cyclic Group, Cosets and Normal Subgroup for 7-mark questions.