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Group Theory II

By Dr. Minirani S   |   Mukesh Patel School of Technology Management and Engineering, NMIMS Deemed University, Mumbai
Learners enrolled: 82
This is a course in abstract algebra, although connections with other fields will be stressed as often as possible. It is a systematic study of the basic structure of groups, finite and infinite. Group theory is one of the great simplifying and unifying ideas in modern mathematics. It was in order to understand the solutions to polynomial equations, but only in the last century has its full significance, as a mathematical formulation of symmetry, been understood.  The course introduces some ideas of group theory, including group automorphisms, application of factor groups to automorphism groups, characteristic subgroups, commutator subgroup and its properties, external and internal direct products, Fundamental theorem of finite abelian groups. We will also discuss in detail the group actions on a set and then go through some of the interesting theorems in group theory such as generalised Cayley’s theorem, Index theorem and Sylow theorems. The theory will be illustrated with examples and problems from leading to its connection with other areas of mathematics. Some of the applications in the mentioned topics will also be discussed. 
Summary
Course Status : Upcoming
Course Type : Core
Language for course content : English
Duration : 16 weeks
Category :
  • Mathematics
Credit Points : 5
Level : Undergraduate
Start Date : 06 Jan 2025
End Date : 25 Apr 2025
Enrollment Ends : 28 Feb 2025
Exam Date : 18 May 2025 IST
NCrF Level   : 5.5
Industry Details : Machine Learning
Exam Shift :

Shift 1

Note: This exam date is subject to change based on seat availability. You can check final exam date on your hall ticket.

Page Visits



Course layout

Week 1
Day 1 Preliminaries
Day 2 Group Automorphisms-I
Day 3 Group Automorphisms-II
Day 4 Group Automorphisms-III
Day 5 Discussion –Live session


Week 2
Day 1 Group Automorphisms-IV
Day 2 Inner Automorphisms-I
Day 3 Inner Automorphisms-II
Day 4 Discussion –Live session

Week 3
Day 1 Automorphism Groups of Cyclic groups
Day 2 Application Of Factor Groups To Automorphism Groups
Day 3 Characteristic Subgroup
Day 4 Discussion –Live session

Week 4
Day 1 Commutator Subgroup
Day 2 External direct product
Day 3 Properties of External direct products-I
Day 4 Properties of External direct products-II
Day 5 Discussion –Live session

Week 5
Day 1 The group of units modulo n as an external direct product
Day 2 Internal direct product-I
Day 3 Internal direct product-II
Day 4 Internal direct product-III
Day 5 Discussion –Live session

Week 6
Day 1 The isomorphism class of abelian groups-I
Day 2 The isomorphism class of abelian groups-II
Day 3 The isomorphism class of abelian groups-III
Day 4 Discussion –Live session

Week 7
Day 1 The isomorphism class of abelian groups-IV
Day 2 Group Actions-I
Day 3 Group Actions-II
Day 4 Discussion –Live session

Week 8
Day 1 Group Actions-III
Day 2 Group Actions-IV
Day 3 Conjugacy-I
Day 4 Discussion –Live session

Week 9
Day 1 Conjugacy-II
Day 2 Conjugacy-III
Day 3 Class equation and Consequences
Day 4 Discussion –Live session

Week 10
Day 1 Class equation and Consequences
Day 2 Conjugacy Classes in Sn And An -Part I
Day 3 Conjugacy Classes in Sn And An -Part II
Day 4 Discussion –Live session


Week 11
Day 1 Sylow Theorems and Applications-Part I
Day 2 Sylow Theorems and Applications-Part II
Day 3 Sylow Theorems and Applications-Part III
Day 4 Discussion –Live session

Week 12
Day 1 Sylow Theorems and Applications-Part IV
Day 2 Consequence of Sylow Theorems-Part I
Day 3 Consequence of Sylow Theorems-Part II
Day 4 Discussion –Live session

Week 13
Day 1 Applications of Sylow Theorems-Part I
Day 2 Applications of Sylow Theorems-Part II
Day 3 Simple Groups
Day 4 Discussion –Live session

Week 14
Day 1 Non-Simplicity Test - Part I
Day 2 Non-Simplicity Test - Part II
Day 3 Non-Simplicity Test - Part III
Day 4 Discussion –Live session

Week 15
Day 1 Non-Simplicity Test - Part IV
Day 2 Non-Simplicity Test - Part V
Day 3 Classification of groups
Day 4 Discussion –Live session

Week 16
Day 1 Some results on Simple groups
Day 2 Simplicity of An - Part I
Day 3 Simplicity of An - Part II
Day 4 Discussion –Live session

Week 17
Day 1 Group Action - Part I
Day 2 Group Action - Part II
Day 3 Sylow Thoerems
Day 4 Discussion –Live session


Books and references

1 I. N. Herstein, Topics in Algebra,  Wiley Eastern Ltd., New Delhi 1975.
2 Nathan Jacobson,  Basic Algebra Vol 1, Hindustan Publishing Corporation (India), Delhi 1993.
3 John B. Fraleigh,  A First Course in Abstract Algebra, Fifth Edition, Addison-Wesley, California, 1999.
4 Vijay K. Khanna, S. K. Bhambri, A Course in Abstract Algebra,  Vikas Publishing House Pvt. Ltd., New Delhi, 1996.
5 N. S. Gopalakrishnan, University Algebra,  Wiley Eastern Limited, New Age International Limited, New Delhi 1995.
6 Surjeet Singh, Qazi Zameeruddin, Modern Algebra,  Vikas Publishing House Pvt. Ltd., New Delhi 1997.
7 Shanti Narayan,  A Text Book of Modern Abstract Algebra,  S. Chand & Co., New Delhi.




Instructor bio

Dr. Minirani S

Mukesh Patel School of Technology Management and Engineering, NMIMS Deemed University, Mumbai
Dr. Minirani S is currently working as an Associate Professor in the Department of Basic Sciences and Humanities in the Mukesh Patel School of technology, Management and Engineering, NMIMS Deemed  to be University, Vile Parle West, Mumbai. Dr. Minirani S has completed her Ph. D. in Mathematics from National Institute of Technology, Calicut and is currently a research guide at NMIMS. She has authored many research articles in national and international journals and a book on fractal geometry. Her areas of interest include Fractal Geometry, Discrete Mathematics and group theory. 

Course certificate

Course Completion  will carry 70% weightage of end term exam and 30% weightage of internal assessments. A minimum 40% in each is required to qualify for the Course Completion Certifcate


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