# Virtual University of Pakistan

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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

 Topic Lecture Resource Page Lecture Number Topic Module No. Lecture # 1 Properties of real numbers 1 Properties of complex numbers 2 Binary Operation 3 Binary Operation 4 Lecture # 2 Bijective maps 5 Inversion Theorem 6 Isomorphic binary structures I 7 Lecture # 3 Isomorphic binary structures II 8 Isomorphic binary structures II 9 Lecture # 4 Groups 10 Examples of Groups 11 Uniqueness of identity and inverse. 12 Example of group 13 Elementary properties of groups 14 - 15 Groups of matrices 16 -17 Lecture # 5 Abelian Groups 18 Abelian Groups 19 Modular Arithmetic 20 Order of a Group 21 Lecture # 6 Finite Groups 22-25 Lecture # 7 Subgroups 26 Examples of subgroups 27 Two Step subgroup test 28-29 One Step subgroup test 30 Lecture # 8 Examples on subgroup test 31 Finite subgroup test 32 Examples on subgroup test 33 Lecture # 9 Cyclic Groups 34 Examples of Cyclic Groups 35-36 Lecture # 10 Elementary properties of Cyclic groups 37-39 Quiz No:1 Lecture # 11 Fundamental Theorem of Cyclic Group 40 Subgroup of finite Cyclic group 41 Theorem of Cyclic Group 42 Lecture # 12 Permutation Groups 43 Examples of Permutation Groups 44-45 Lecture # 13 Theorem on permutation groups 46 Cayley''s Theorem 47 Examples of Permutation Groups 48 Lecture # 14 Orbits 50-51 Cycles 52 Disjoint Cycles 53 Lecture # 15 Cycle Decomposition 54 Parity of permutations 55 Alternating Group 56 Lecture # 16 Direct product 57-59 Lecture # 17 Direct product 60-62 Finitely generated abelian groups 63 Lecture # 18 Applications 64-65 Cosets 66 Partition of group 67 Lecture # 19 Examples of Costes 68-69 Properties of costes 70-72 Lecture # 20 Lagrange''s Theorem 73 Applications of Lagrange''s theorem 74 Indices of subgroups 75-76 Lecture # 21 Converse of Lagrange''s Theorem 77-78 Homomorphism of Groups 79-83 Lecture # 22 Properties of homomorphism 84-85 Normal Subgroups 86-89 Grand Quiz Lecture # 23 Morphism Theorem for groups 90 Application of Morphism theorem 91 properties of homomorphism 92 Normality of kernel of homomorphism 93 Lecture # 24 Example of Normal group 94-95 Factor group 96 Cosets multiplcation & Normality 97 Lecture # 25 Examples on kernel of homomorphisms 98-101 Lecture # 26 Examples on kernel of homomorphisms 102-103 Examples of group homomorphism 104 -105 Lecture # 27 Factor group from homomorphism 106-109 Lecture # 28 Factor groups from Normal subgroups 110-111 Kernel of an injective homomorphism 112 Factor groups from Normal subgroups 113 Lecture # 29 Example of Morphism theorem of groups 114 Normal groups and Inner Automorphism 115-119 Lecture # 30 Theorem on Factor group 120 Example on Factor group 121 Factor group computations 122-123 Lecture # 31 Factor group computations 124-126 Lecture # 32 Factor group computations 127-130 Lecture # 33 Simple Group 131-134 Maximal Normal Subgroups 135 The Centre subgroup 136 Lecture # 34 Example of the Centre subgroup 137-138 Commutator subgroup 139 Lecture # 35 Generating set 140-143 Lecture # 36 Commutator subgroup 144-146 Lecture # 37 Automorphisms 147-149 Lecture # 38 Examples on Automorphisms 150-152 Lecture # 39 Group Action on set 153-157 Lecture # 40 Stablizer 158-159 Orbits 160 Lecture # 41 Conjugacy and G-sets 161-162 Isomorphism Theorem 163-164 Lecture # 42 Second Isomorphism Theorem 165-167 Third Isomorphism Theorem 168-170 Lecture # 43 Sylow Theorems 171-174 Lecture # 44 First Sylow Theorems 175 Second Sylow Theorems 176 Third Sylow Theorems 177 Sylow Theorems 178 Lecture # 45 Application of Sylow Theorems 179-181 Final Paper