Group Theory I

The course “Group Theory I” is proposed for B.Sc. (Hons) Mathematics. The course starts with the overview of binary operations on a set, the definition of groups, cyclic groups, groups of symmetries of a square, the external direct product of groups, normal subgroups, homomorphism, the kernel of homomorphisms, Isomorphisms, and the three isomorphism theorems

Weeks Weekly Lecture Topics (Module Titles)

1 Day 1 Module 1: Binary Operations Day 2 Module 2: Introduction to Groups
Day 3 Module 3: More Examples to Groups Day 4 Assignment Day 5 Quiz

2 Day 1 Module 4 : Elementary properties of Groups Day 2 Module 5 : Subgroups
Day 3 Module 6 : Cyclic Groups
Day 4 Assignment Day 5 Quiz
3 Day 1 Module 7 : Properties of Cyclic Groups Day 2 Module 8 : Classification of Subgroups of Cyclic Groups
Day 3 Module 9 : Problems on subgroups of cyclic group
Day 4 Assignment Day 5 Quiz
4 Day 1 Module 10 : Order of an element Day 2 Module 11 : : Permutation Groups
Day 3 Module 12 Symmetric Group S_n Day 4 Assignment Day 5 Quiz
5 Day 1 Module 13 : Orbits and Cycles Day 2 Module 14 : On Cyclic Permutations
Day 3 Module 15 : Permutation as product of Transpositions Day 4 Assignment Day 5 Quiz
6 Day 1 Module 16 : Alternating Groups A_n Day 2 Module 17 : Generating Sets of S-n and A_n
Day 3 Module 18 : Conjugate Elements,Conjugacy Class Day 4 Assignment Day 5 Quiz
7 Day 1 Module 19 : Conjugacy in S_n Day 2 Module 20 : The Dihedral Group
Day 3 Module 21 : Quaternion Group
Day 4 Assignment Day 5 Quiz
8 Day 1 Module 22 : Some Problems of Permutation Group Day 2 Module 23: Cosets
Day 3 Module 24 : Lagrange's Theorem Day 4 Assignment Day 5 Quiz
9 Day 1 Module 25 : Fermat's Theorem Day 2 Module 26 : External Direct Product -Part I
Day 3 Module 27: External Direct Product -Part II
Day 4 Assignment Day 5 Quiz
10 Day 1 Module 28 : Group Homomorphism -Part I Day 2 Module 29 : Properties of Homomorphism
Day 3 Module 30 : Group Homomorphism -Part II Day 4 Assignment
Day 5 Quiz
11 Day 1 Module 31 : Normal Subgroups Day 2 Module 32 : Kernel of a Homomorphism-Part I
Day 3 Module 33: Kernel of a Homomorphism-Part II Day 4 Assignment Day 5 Quiz
12 Day 1 Module 34 : Index of a Subgroup Day 2 Module 35 : Factor Groups-Part I
Day 3 Module 36 : Factor Groups-Part II Day 4 Assignment Day 5 Quiz
13 Day 1 Module 37 : Examples of Normal Subgroups Day 2 Module 38 : Problems in Cosets and Factor Groups
Day 3 Module 39 : Cauchy's Theorem Day 4 Assignment Day 5 Quiz
14 Day 1 Module 40 : Group Isomorphism Day 2 Module 41 : Properties of Isomorphism
Day 3 Module 42 : The First Isomorphism Theorem Day 4 Assignment Day 5 Quiz
15 Day 1 Module 43 : Prerequisites for Second Isomorphism Theorem Day 2 Module 44 : Second Isomorphism Theorem
Day 3 Module 45 : Third Isomorphism Theorem Day 4 Assignment Day 5 Quiz
16 Day 1 Module 46 : Cayley's Theorem Day 2 Module 47 : Product of Subgroups
Day 3 Module 48 : Center of a Group Day 4 Assignment Day 5 Quiz
17 Day 1 Module 49 : Centralizer of an element Day 2 Module 50 : Normalizer of a subgroup
Day 3 Module 51: Commutator Subgroup Day 4 Assignment Day 5 Quiz