Modern Algebra

Teacher

Dr. Swarnalata Jena

Category

Core Courses

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Course Name : Modern Algebra

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 Plan

Session 1

Introduction to group theory

https://www.youtube.com/watch?v=zkADn-9wEgc

Session 2

Algebraic structure of group

https://www.youtube.com/watch?v=RqX3ZPnANrU

Session 4

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

https://www.youtube.com/watch?v=j01ETdSI9zY

Session 9

Session 11

Relation between cyclic group and abelian group

https://www.youtube.com/watch?v=kFSF7E2wAZA

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

https://www.youtube.com/watch?v=HJ3QeZrrLdA

Session 15

Session 16

Session 17

Index of a subgroup H in G

https://www.youtube.com/watch?v=x6kByMOUe8g

Session 18

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

https://www.youtube.com/watch?v=1jPaWx7jnIg

Session 22

Composition of two permutations

https://www.youtube.com/watch?v=Q8HaZsAbgeQ

Session 23

Session 24

Session 25

Session 26

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

https://www.youtube.com/watch?v=YOO94VaYS8I

Session 31

Session 32

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

https://www.youtube.com/watch?v=C2CgkRaLf3s

Session 38

Session 39

Properties of Homomorphism

https://www.youtube.com/watch?v=b5gchgQNpWM

Session 40

Connection between homomorphism and normal subgroup

https://www.youtube.com/watch?v=t3JR0VGOY30

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

https://www.youtube.com/watch?v=hjh1D_Psox0

Session 45

Session 46

Session 47

Project-17 First, second and third Isomorphism theorems

Session 48

Project-18 Uses of Cayleys Theorem

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