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MTH633 : Group Theory

Course Overview

Course Synopsis

Course Synopsis The following topics are the focus of this course: Set, binary operations, groups, order of group and elements, subgroups, Abelian group, Cyclic group, permutation group, orbits, direct product, cosets, Lagrange’s theorem, Homomorphism of groups, Factor groups, First Isomorphism Theorem, Second Isomorphism Theorem, Third Isomorphism Theorem, Group actions, First Sylow’s Theorem, Second Sylow’s Theorem, Third Sylow’s Theorem,

Course Learning Outcomes

At the end of the course, you should be able to :

  • students will be able to understand Binary Operations
  • Determine whether a set is a group.
  • Define and determine the subgroups of a group
  • Understand the order of a group and its elements
  • Understand the cyclic group
  • Understand the group of permutations
  • understand and apply Lagrange''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''s theorem
  • determine the group isomorphisms
  • determine group actions
  • understand and apply Sylow's theorems

Course Calendar

Lecture NumberTopicModule No.
Lecture # 1Properties of real numbers1
Properties of complex numbers2
Binary Operation3
Binary Operation4
Lecture # 2Bijective maps5
Inversion Theorem6
Isomorphic binary structures I7
Lecture # 3Isomorphic binary structures II8
Isomorphic binary structures II9
Lecture # 4Groups10
Examples of Groups11
Uniqueness of identity and inverse.12
Example of group13
Elementary properties of groups14 - 15
Groups of matrices16 -17
Lecture # 5Abelian Groups18
Abelian Groups19
Modular Arithmetic20
Order of a Group21
Lecture # 6Finite Groups22-25
Lecture # 7Subgroups26
Examples of subgroups27
Two Step subgroup test28-29
One Step subgroup test30
Lecture # 8Examples on subgroup test31
Finite subgroup test32
Examples on subgroup test33
Lecture # 9Cyclic Groups34
Examples of Cyclic Groups35-36
Lecture # 10Elementary properties of Cyclic groups37-39
Quiz No:1
Lecture # 11Fundamental Theorem of Cyclic Group40
Subgroup of finite Cyclic group41
Theorem of Cyclic Group42
Lecture # 12Permutation Groups43
Examples of Permutation Groups44-45
Lecture # 13Theorem on permutation groups46
Cayley''s Theorem47
Examples of Permutation Groups48
Lecture # 14Orbits50-51
Disjoint Cycles53
Lecture # 15Cycle Decomposition54
Parity of permutations55
Alternating Group56
Lecture # 16Direct product57-59
Lecture # 17Direct product60-62
Finitely generated abelian groups63
Lecture # 18Applications64-65
Partition of group67
Lecture # 19Examples of Costes68-69
Properties of costes70-72
Lecture # 20Lagrange''s Theorem73
Applications of Lagrange''s theorem74
Indices of subgroups75-76
Lecture # 21Converse of Lagrange''s Theorem77-78
Homomorphism of Groups79-83
Lecture # 22Properties of homomorphism84-85
Normal Subgroups86-89
Grand Quiz
Lecture # 23Morphism Theorem for groups90
Application of Morphism theorem91
properties of homomorphism92
Normality of kernel of homomorphism93
Lecture # 24Example of Normal group94-95
Factor group96
Cosets multiplcation & Normality97
Lecture # 25Examples on kernel of homomorphisms98-101
Lecture # 26Examples on kernel of homomorphisms102-103
Examples of group homomorphism104 -105
Lecture # 27Factor group from homomorphism106-109
Lecture # 28Factor groups from Normal subgroups110-111
Kernel of an injective homomorphism112
Factor groups from Normal subgroups113
Lecture # 29Example of Morphism theorem of groups114
Normal groups and Inner Automorphism115-119
Lecture # 30Theorem on Factor group120
Example on Factor group121
Factor group computations122-123
Lecture # 31Factor group computations124-126
Lecture # 32Factor group computations127-130
Lecture # 33Simple Group131-134
Maximal Normal Subgroups135
The Centre subgroup136
Lecture # 34Example of the Centre subgroup137-138
Commutator subgroup139
Lecture # 35
Generating set140-143
Lecture # 36Commutator subgroup144-146
Lecture # 37Automorphisms147-149
Lecture # 38Examples on Automorphisms150-152
Lecture # 39Group Action on set153-157
Lecture # 40Stablizer158-159
Lecture # 41Conjugacy and G-sets161-162
Isomorphism Theorem163-164
Lecture # 42Second Isomorphism Theorem165-167
Third Isomorphism Theorem168-170
Lecture # 43Sylow Theorems171-174
Lecture # 44First Sylow Theorems175
Second Sylow Theorems176
Third Sylow Theorems177
Sylow Theorems178
Lecture # 45Application of Sylow Theorems179-181
Final Paper
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