Overview

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.

Syllabus

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