# Modern Algebra

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# Code(Credit) : CUTM1516(3-0-3)

## Course Objectives

• A major objective is to introduce students to the language and precision of modern algebra. This means that the course will be proof-based, in the sense that students will be expected to understand, construct, and write proofs.
• A challenge for all students of mathematics is to balance the understanding with the communication. There is a tendency to think you are finished once you see why a mathematical statement is true or false.

In fact you are just half-way there because constructing a legitimate proof involves different skills and expertise than the discovery part of the process. In this course both angles of problem-solving will be stressed.

## Learning Outcomes

Upon successful completion of this course, students will be able to:

• Effectively write abstract mathematical proofs in a clear and logical manner.
• Locate and use theorems to solve problems in number theory and theory of polynomials over a field.
• Demonstrate ability to think critically by interpreting theorems and relating results to problems in other mathematical disciplines.
• Demonstrate ability to think critically by recognizing patterns and principles of algebra and relating them to the number system.

Work effectively with others to discuss homework problems put on the board.

## Course Syllabus

Module-I

Definition and examples of groups, Subgroups and examples of subgroups.

Project-1 Collection of abelian groups using different properties.

Project-2 Report on multiplication modulo and addition modulo groups.

Project-3 Finding elements of Un(K)

Module-II

Lagrange’s Theorem and Consequences, Fermat’s little theorem, Cyclic groups of Group G.

Project-4 Finding order of a subgroup using Lagrange’s Theorem

Project-5 Uses of Fermat’s little theorem

Module-III

Classification of Subgroups of Cyclic group, Cosets and Properties of Cosets

Project-6 List of the elements of the groups < n > in Zm

Project-7 Index of a subgroup H in G

Module-IV

Permutation Groups

Project-8 A group model of A4

Project-9 Digit scheme based on D5

Project-10 Rotation of Tetrahedron

Module-V

Application of cosets to permutation groups, Normal subgroups

Project-11 Collections of application of cosets to different groups

Project-12 Application to public key cryptography

Project-13 Reports on Stabilizer point and Orbit point

Module-VI

Quotient groups, Group Homomorphism, Properties of Homomorphism

Project-14 Reports on Quotient groups

Project-15 Properties of Homomorphism with examples.

Project-16 Collection on examples of group Homomorphism.

Module-VII

Isomorphism: Definition and examples, Cayleys Theorem

Project-17 First, second and third Isomorphism theorems

Project-18 Uses of Cayleys Theorem

Text Book:

1. Joseph A. Gallian, Contemporary Abstract Algebra (4th Edn.), Narosa Publishing House, New Delhi.

Chapters: I, II, III, IV, V, VI VII, IX

Reference Books:

1. University Algebra – N.S. Gopalkrishna (Wiley Eastern)
2. Modern Algebra – Vatsa and Vatsa (New Age International)
3. Topics in algebra - I. N. Herstein (Vikas Pub. House)

## Session 1

Introduction to group theory

## Session 2

Algebraic structure of group

## Session 5

Project-1 Collection of abelian groups using different properties.

## Session 6

Project-2 Report on multiplication modulo and addition modulo groups.

## Session 7

Project-3 Finding elements of Un(K)

## Session 8

Lagrange’s Theorem and Consequences

## Session 11

Relation between cyclic group and abelian group

## Session 12

Project-4 Finding order of a subgroup using Lagrange’s Theorem

## Session 13

Project-5 Uses of Fermat’s little theorem

## Session 14

Classification of subgroups of Cyclic group

## Session 17

Index of a subgroup H in G

## Session 19

Project-6 List of the elements of the groups < n > in Zm

## Session 20

Project-7 Index of a subgroup H in G

## Session 21

Permutation Group of a set

## Session 22

Composition of two permutations

## Session 27

Project-8 A group model of A4

## Session 28

Project-9 Digit scheme based on D5

## Session 29

Project-10 Rotation of Tetrahedron

## Session 30

Application of cosets to permutation groups

## Session 33

Project-11 Collections of application of cosets to different groups

## Session 34

Project-12 Application to public key cryptography

## Session 35

Project-13 Reports on Stabilizer point and Orbit point

## Session 37

Application of Quotient groups

## Session 39

Properties of Homomorphism

## Session 40

Connection between homomorphism and normal subgroup

## Session 41

Project-14 Reports on Quotient groups

## Session 42

Project-15  Project on properties of Homomorphism with examples.

## Session 43

Project-16 Collection on examples of group Homomorphism.

## Session 44

Definition and examples of Isomorphism

## Session 47

Project-17 First, second and third Isomorphism theorems

## Session 48

Project-18 Uses of Cayleys Theorem