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.

Course Outcomes

COs Course outcomes Mapping Cos with POs (High-3, Medium-2, Low-1)
 

CO1

 

 

Effectively write abstract mathematical proofs in a clear and logical manner.

 

 

PO1(3), PO2(2)

CO2  

Locate and use theorems to solve problems in number theory and theory of polynomials over a field.

 

PO2(2)
CO3  

Demonstrate ability to think critically by interpreting theorems and relating results to problems in other mathematical disciplines.

 

PO4(2), PO9(2)
CO4  

Demonstrate ability to think critically by recognizing patterns and principles of algebra and relating them to the number system.

 

PO1(2), PO2(2),

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 & 6

Project-1 Collection of abelian groups using different properties.

Session 7 & 8

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

Session 9 & 10

Project-3 Finding elements of Un(K)

Session 11

Lagrange’s Theorem and Consequences

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

Session 12

Session 14

Relation between cyclic group and abelian group

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

Session 15 & 16

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

Session 17 & 18

Project-5 Uses of Fermat’s little theorem

Session 19

Classification of subgroups of Cyclic group

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

Session 20

Session 21

Session 22

Index of a subgroup H in G

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

Session 23

Session 24 & 25

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

Session 26 & 27

Project-7 Index of a subgroup H in G

Session 28

Permutation Group of a set

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

Session 29

Composition of two permutations

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

Session 30

Session 31

Session 32

Session 33

Session 34 & 35

Project-8 A group model of A4

Session 36 & 37

Project-9 Digit scheme based on D5

Session 38 & 39

Project-10 Rotation of Tetrahedron

Session 40

Application of cosets to permutation groups

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

Session 41

Session 42

Session 43 & 44

Project-11 Collections of application of cosets to different groups

Session 45 & 46

Project-12 Application to public key cryptography

Session 47 & 48

Project-13 Reports on Stabilizer point and Orbit point

Session 50

Application of Quotient groups

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

Session 51

Session 52

Properties of Homomorphism

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

Session 53

Connection between homomorphism and normal subgroup

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

Session 54 & 55

Project-14 Reports on Quotient groups

Session 56 & 57

Project-15  Project on properties of Homomorphism with examples.

Session 58 & 59

Project-16 Collection on examples of group Homomorphism.

Session 60

Definition and examples of Isomorphism

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

Session 61

Session 62

Session 63 & 64

Project-17 First, second and third Isomorphism theorems

Session 65 & 66

Project-18 Uses of Cayleys Theorem

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