Course Overview
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Course Synopsis
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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
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Course Learning Outcomes
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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
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Course Calendar
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Week 01
1
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Properties of real numbers
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2
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Properties of complex numbers
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7
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Isomorphic binary structures I
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8
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Isomorphic binary structures II
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9
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Isomorphic binary structures III
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Week 02
12
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Uniqueness of identity and inverse.
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14
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Elementary properties of groups I
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15
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Elementary properties of groups II
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Week 03
28
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Two Step subgroup test I
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29
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Two Step subgroup test II
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30
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One Step subgroup test
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31
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Examples on subgroup test
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33
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Examples on subgroup test II
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35
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Examples of Cyclic Groups I
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36
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Examples of Cyclic Groups II
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Week 04
37
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Elementary properties of Cyclic groups I
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38
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Elementary properties of Cyclic groups II
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39
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Elementary properties of Cyclic groups III
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40
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Fundamental Theorem of Cyclic Group
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41
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Subgroup of finite Cyclic group
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42
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Theorem of Cyclic Group
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44
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Examples of Permutation Groups I
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45
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Examples of Permutation Groups II
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Week 05
46
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Theorem on permutation groups
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48
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Examples of Permutation Groups
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55
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Parity of permutations
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Week 06
63
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Finitely generated abelian groups
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Week 07
70
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Properties of cosets I
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71
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Properties of cosets II
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72
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Properties of cosets III
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74
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Applications of Lagrange's theorem
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75
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Indices of subgroups I
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76
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Indices of subgroups II
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77
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Converse of Lagrange's Theorem I
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78
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Converse of Lagrange's Theorem II
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79
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Homomorphism of Groups I
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80
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Homomorphism of Groups II
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81
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Homomorphism of Groups III
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82
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Homomorphism of Groups IV
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83
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Homomorphism of Groups V
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Week 08
84
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Properties of homomorphism I
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85
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Properties of homomorphism II
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Week 09
90
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Morphism Theorem for groups
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91
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Application of Morphism theorem
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92
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properties of homomorphism
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93
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Normality of kernel of homomorphism
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94
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Example of Normal group I
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95
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Example of Normal group II
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97
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Cosets multiplcation & Normality
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98
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Examples on kernel of homomorphisms I
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99
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Examples on kernel of homomorphisms II
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100
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Examples on kernel of homomorphisms III
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101
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Examples on kernel of homomorphisms IV
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Week 10
102
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Examples on kernel of homomorphisms V
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103
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Examples on kernel of homomorphisms VI
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104
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Examples of group homomorphism I
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105
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Examples of group homomorphism II
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106
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Factor group from homomorphism I
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107
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Factor group from homomorphism II
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108
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Factor group from homomorphism III
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109
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Factor group from homomorphism IV
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110
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Factor groups from Normal subgroups I
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111
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Factor groups from Normal subgroups II
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112
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Kernel of an injective homomorphism
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113
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Factor groups from Normal subgroups
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114
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Example of Morphism theorem of groups
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Week 11
115
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Normal groups and Inner Automorphism I
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116
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Normal groups and Inner Automorphism II
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117
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Normal groups and Inner Automorphism III
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118
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Normal groups and Inner Automorphism IV
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119
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Normal groups and Inner Automorphism V
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120
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Theorem on Factor group
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121
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Example on Factor group
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122
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Factor group computations I
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123
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Factor group computationsII
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124
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Factor group computations III
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125
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Factor group computations IV
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126
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Factor group computations V
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Week 12
127
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Factor group computations VI
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128
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Factor group computations VII
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129
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Factor group computations VIII
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130
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Factor group computations IX
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135
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Maximal Normal Subgroups
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137
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Example of the Centre subgroup I
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138
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Example of the Centre subgroup II
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Week 13
144
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Commutator subgroup I
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145
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Commutator subgroup II
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146
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Commutator subgroup III
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Week 14
150
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Examples on Automorphisms I
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151
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Examples on Automorphisms II
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152
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Examples on Automorphisms III
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153
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Group Action on set I
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154
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Group Action on set II
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155
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Group Action on set III
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156
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Group Action on set IV
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157
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Group Action on set V
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Week 15
161
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Conjugacy and G-sets I
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162
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Conjugacy and G-sets II
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163
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Isomorphism Theorem I
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164
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Isomorphism Theorem II
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165
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Second Isomorphism Theorem I
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166
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Second Isomorphism Theorem II
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167
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Second Isomorphism Theorem III
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168
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Third Isomorphism Theorem I
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169
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Third Isomorphism Theorem II
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170
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Third Isomorphism Theorem III
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Week 16
176
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Second Sylow Theorems
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179
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Application of Sylow Theorems I
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180
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Application of Sylow Theorems II
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181
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Application of Sylow Theorems III
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