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MTH633 : Group Theory

Course Overview

Course Synopsis

Course Synopsisrn rnThe following topics are the focus of this course rnrnSet binary operations groups order of group and elements subgroups Abelian group Cyclic group permutation group orbits direct product cosets Lagranges theorem Homomorphism of groups Factor groups First Isomorphism Theorem Second Isomorphism Theorem Third Isomorphism Theorem Group actions First Sylows Theorem Second Sylows Theorem Third Sylows 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 Lagranges theorem
  • determine the group isomorphisms
  • determine group actions
  • understand and apply Sylow39s theorems

Course Calendar

1 Properties of real numbers
2 Properties of complex numbers
3 Binary Operation 1
4 Binary Operation 2
5 Bijective maps
6 Inversion Theorem
7 Isomorphic binary structures I
8 Isomorphic binary structures II
9 Isomorphic binary structures III

10 Groups
11 Examples of Groups
12 Uniqueness of identity and inverse.
13 Example of group
14 Elementary properties of groups I
15 Elementary properties of groups II
16 Groups of matrices I
17 Groups of matrices II
18 Abelian Groups
19 Abelian Groups II
20 Modular Arithmetic
21 Order of a Group
22 Finite Groups
23 Finite Groups II
24 Finite Groups III
25 Finite Groups IV

26 Subgroups
27 Examples of subgroups
28 Two Step subgroup test I
29 Two Step subgroup test II
30 One Step subgroup test
31 Examples on subgroup test
32 Finite subgroup test
33 Examples on subgroup test II
34 Cyclic Groups
35 Examples of Cyclic Groups I
36 Examples of Cyclic Groups II

37 Elementary properties of Cyclic groups I
38 Elementary properties of Cyclic groups II
39 Elementary properties of Cyclic groups III
40 Fundamental Theorem of Cyclic Group
41 Subgroup of finite Cyclic group
42 Theorem of Cyclic Group
43 Permutation Groups
44 Examples of Permutation Groups I
45 Examples of Permutation Groups II
Assignment No.1 Module 1-45

46 Theorem on permutation groups
47 Cayley's Theorem
48 Examples of Permutation Groups
49 Orbits
50 Orbits I
51 Orbits II
52 Cycles
53 Disjoint Cycles
54 Cycle Decomposition
55 Parity of permutations
56 Alternating Group

57 Direct product I
58 Direct product II
59 Direct product III
Quiz No.1 Module 1-33
60 Direct product IV
61 Direct product V
62 Direct product VI
63 Finitely generated abelian groups
64 Applications I
65 Applications II
66 Cosets
67 Partition of group

68 Examples of Costes I
69 Examples of Costes II
70 Properties of cosets I
71 Properties of cosets II
72 Properties of cosets III
73 Lagrange's Theorem
74 Applications of Lagrange's theorem
75 Indices of subgroups I
76 Indices of subgroups II
Quiz No.2 Module 34-76
77 Converse of Lagrange's Theorem I
78 Converse of Lagrange's Theorem II
79 Homomorphism of Groups I
80 Homomorphism of Groups II
81 Homomorphism of Groups III
82 Homomorphism of Groups IV
83 Homomorphism of Groups V

84 Properties of homomorphism I
85 Properties of homomorphism II
86 Normal Subgroups I
87 Normal Subgroups II
88 Normal Subgroups III
89 Normal Subgroups IV
Mid term Exams Module 1-89

90 Morphism Theorem for groups
91 Application of Morphism theorem
92 properties of homomorphism
93 Normality of kernel of homomorphism
94 Example of Normal group I
95 Example of Normal group II
96 Factor group
97 Cosets multiplcation & Normality
98 Examples on kernel of homomorphisms I
99 Examples on kernel of homomorphisms II
100 Examples on kernel of homomorphisms III
101 Examples on kernel of homomorphisms IV

102 Examples on kernel of homomorphisms V
103 Examples on kernel of homomorphisms VI
104 Examples of group homomorphism I
105 Examples of group homomorphism II
106 Factor group from homomorphism I
107 Factor group from homomorphism II
108 Factor group from homomorphism III
109 Factor group from homomorphism IV
110 Factor groups from Normal subgroups I
111 Factor groups from Normal subgroups II
112 Kernel of an injective homomorphism
113 Factor groups from Normal subgroups
114 Example of Morphism theorem of groups

115 Normal groups and Inner Automorphism I
116 Normal groups and Inner Automorphism II
117 Normal groups and Inner Automorphism III
118 Normal groups and Inner Automorphism IV
119 Normal groups and Inner Automorphism V
120 Theorem on Factor group
121 Example on Factor group
122 Factor group computations I
123 Factor group computationsII
124 Factor group computations III
125 Factor group computations IV
126 Factor group computations V
Quiz No.3 Module 90-126

127 Factor group computations VI
128 Factor group computations VII
129 Factor group computations VIII
130 Factor group computations IX
131 Simple Group I
132 Simple Group II
133 Simple Group III
134 Simple Group IV
135 Maximal Normal Subgroups
136 The Centre subgroup
137 Example of the Centre subgroup I
138 Example of the Centre subgroup II
139 Commutator subgroup

140 Generating set I
141 Generating set II
142 Generating set III
143 Generating set IV
144 Commutator subgroup I
145 Commutator subgroup II
146 Commutator subgroup III
Quiz No.4 Module 127-146
147 Automorphisms I
148 Automorphisms II
149 Automorphisms III

150 Examples on Automorphisms I
151 Examples on Automorphisms II
152 Examples on Automorphisms III
153 Group Action on set I
154 Group Action on set II
155 Group Action on set III
156 Group Action on set IV
157 Group Action on set V
158 Stablizer I
159 Stablizer II
160 Orbits III

161 Conjugacy and G-sets I
162 Conjugacy and G-sets II
163 Isomorphism Theorem I
164 Isomorphism Theorem II
165 Second Isomorphism Theorem I
166 Second Isomorphism Theorem II
167 Second Isomorphism Theorem III
168 Third Isomorphism Theorem I
169 Third Isomorphism Theorem II
170 Third Isomorphism Theorem III
171 Sylow Theorems I
172 Sylow Theorems II
173 Sylow Theorems III
174 Sylow Theorems IV

175 First Sylow Theorems
176 Second Sylow Theorems
177 Third Sylow Theorems
178 Sylow Theorems V
179 Application of Sylow Theorems I
180 Application of Sylow Theorems II
181 Application of Sylow Theorems III
End Semester Exams

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