Lectures : Monday, Wednesday and Thursday at 10:00 am
Tutorial: TBA
Minor Examination - 20%
Major Examination - 40%
Quizzes - 25%
Tutorials - 5%
Assignments - 10%
Practice Problems sheets will be provided every week and same will be discussed during the Tutorial.
A student is expected to have full attendance in the course unless student takes leave of absence for valid medical or bonafide reasons. In any case, at least 75% attendance in the course is mandatory.
To train the student in the domain of Abstract Algebra.
To give sufficient knowledge of abstract algebra, required to do advanced courses in pure and applied mathematics.
The student will be able to
1. precisely define terms like group, ring, field, subgroup, normal subgroup, ideal, and quotient group.
2. identify and provide examples of various algebraic structures, such as cyclic groups, Abelian Groups, Dihedral Groups, Symmetric and Alternating Groups, General Linear Groups, and rings.
3. apply the theory to solve a variety of problems, including finding the order of elements in groups, determining the structure of finite abelian groups, to determine if a given group is simple.
4. apply the concept of group actions in counting techniques.
5. develop abstract reasoning and thinking such as recognizing patterns, symmetries and making generalizations.
Group Theory: Definition of groups, Examples and basic properties of groups, Cyclic groups, Abelian Groups, Dihedral Groups, Symmetric and Alternating Groups, General Linear Groups, Subgroups. [9 lectures]
Normal Subgroups, Quotient groups, Homomorphisms, Kernel and Image, Isomorphism theorems. Automorphism groups. Simple groups and Simplicity of Alternating groups. Direct Products, Statement
of structure theorem of finite abelian groups. [9 lectures]
Group actions, Examples of group action, Conjugacy Classes, Centralizer, Normalizer, Class Equation, Cayley’s Theorem, Lagrange’s Theorem, Cauchy’s Theorem, The Sylow Theorems (without proof) and their applications. [12 lectures]
Ring Theory: Definition of Rings, Examples and basic properties of rings, Subrings and Characteristics of a ring, Ideals, Sum and direct sum of ideals, Definition of Fields, Examples of finite and infinite fields. Applications in Cryptography and Coding Theory [9 lectures]
1. J. A. Gallian, Contemporary Abstract algebra, Fourth Edition, Narosa Publishing House.
2. P. B. Bhattacharya, S.K. Jain, S. R. Nagpaul, Basic Abstract Algebra, Second Edition, Cambridge University Press, 1994
1. M. Artin, Algebra, Pearson Education, 2011.
2. I. N. Herstein, Topics in Algebra, New York: Wiley, 1975.
D. S. Dummit and R. M. Foote: Abstract Algebra, 2nd Edition, John Wiley, 2002.
5,7,8 January : Introduction to the Course, Symmetries and Combining symmteries.
12,14,15 January : Definition of Group, Group Axioms and Examples
19, 21, 22 January : More Group Examples and Subgroups (Quiz - I)
28, 29 January : Normal subgroups and Quotient groups
2,4,5 February : Homomorphisms and Isomorphism Theorems
9,11,12 February: Automorphism Groups and Introduction to Simple Groups (Quiz -II)
16 February : Alternating Groups are simple groups for n > 4.
2,5,7 March : Group Actions and their Examples
9,11 March : Conjugacy Classes and Centralizers, Class Equations
18,19 March : Cayley's Theorem, Lagrange's Theorem (Quiz-III)
23, 25, 26 March : Cauchy's Theorem and Sylow's Theorems (Without Proof) with applications
30 March : Definition of Ring and Examples.
1,2 April : Subrings and Characterstic of a ring.
6,8,9 April : Ideals and Operations on Ideals. (Quiz-IV)
13-16 April : Applications in Cryptography and Coding Theory.
20 April : Definition of Fields, Examples and Finite Fields.