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01 Introduction
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02 Introduction to sets
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03 Subsets
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04 Our first proof
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05 Products sets and mappings
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06 More about mappings
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08 Exercise problem
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09 Exercise problem
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10 Relations (still with the not-so-exciting-stuff)
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11 Reflexive, symmetric, and transitive properties of relations
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12 Equivalence relations
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13 Equivalence sets
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14 Ordering of sets
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15 Properties of partially ordered sets
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16 You have made it to the first exciting video Operations
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17 Examples of binary operations
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18 Types of binary operations
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19 Defining the types of binary operations
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20 The identity element
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21 The inverse element
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22 Combinations of binary operations
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23 Algebraic system isomorphism
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Definition of a group Lesson 24
24
Cancellation law Lesson 25
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Group equation examples Lesson 26
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Groups that commute Lesson 27
27
Example problems with inverses Lesson 28
28
Integers modulo n
29
Groups in abstract algebra examples
30
Subgroups abstract algebra
31
Cyclic groups and generators
32
Symmetric groups
33
Homomorphisms in abstract algebra
34
Homomorphisms in abstract algebra examples
35
Isomorphisms in abstract algebra
36
Proof that Cayley table row and column entries are unique and complete
37
Cayley theorem proof
38
Cosets in abstract algebra
39
Cosets and equivalence class proof
40
Coset example
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Coset deeper insights
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Normal subgroups
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Cosets generated by elements of cosets
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Lagrange theorem
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Quotient groups
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Quotient group example
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Product groups
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Product group example
49
Surjective homomorphisms in abstract algebra
50
Kernel of a group homomorphism
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First theorem of isomorphisms
52
Group actions in abstract algebra
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Group action proofs in abstract algebra
54
Group action examples
55
Orbit of a set in abstract algebra
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Stabilizer in abstract algebra
57
Example of a stabilizer in abstract algebra
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Orbit stabilizer theorem
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Dihedral groups in abstract algebra
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Dihedral group example
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Center of a group in abstract algebra
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Centralizer of an element in the dihedral group of order 6
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Centralizer of a set in a group
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Normalizer of a set in a group
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Conjugation of a group
66
Conjugacy classes of a group
67
Class equation of a group
68
Cycle types and conjugacy classes of the symmetric group
69
Group automorphisms in abstract algebra
70
Group automorphism example
71
How to use Plotly in a Google Colaboratory notebook
Description:
Delve into the fundamental concepts of abstract algebra through this comprehensive 10-hour course. Begin with an introduction to sets, subsets, and proofs before exploring product sets, mappings, and relations. Progress to more advanced topics such as binary operations, algebraic system isomorphism, and group theory. Examine various types of groups, including cyclic, symmetric, and dihedral groups, as well as concepts like homomorphisms, isomorphisms, and cosets. Investigate important theorems such as Lagrange's theorem and the First Isomorphism Theorem. Explore group actions, orbits, stabilizers, and conjugacy classes. Conclude with an introduction to group automorphisms and learn how to use Plotly in a Google Colaboratory notebook for data visualization.

Abstract Algebra

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01 Introduction