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:
- Joseph A. Gallian, Contemporary Abstract Algebra (4th Edn.), Narosa Publishing House, New Delhi.
Chapters: I, II, III, IV, V, VI VII, IX
Reference Books:
- University Algebra – N.S. Gopalkrishna (Wiley Eastern)
- Modern Algebra – Vatsa and Vatsa (New Age International)
- Topics in algebra - I. N. Herstein (Vikas Pub. House)
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 14
Relation between cyclic group and abelian group
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 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 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 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 53
Connection between homomorphism and normal subgroup
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 63 & 64
Project-17 First, second and third Isomorphism theorems
Session 65 & 66
Project-18 Uses of Cayleys Theorem
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