This course on abstract algebra aims to teach students the following learning outcomes and goals: understanding the fundamental concepts of abstract algebra, such as sets, relations, binary operations, groups, subgroups, homomorphisms, isomorphisms, and group actions.
Students will learn about various tools and skills, including defining binary operations, identifying group properties, solving group equations, working with cosets, applying Lagrange's theorem, understanding quotient groups, and exploring group automorphisms.
The teaching method of the course involves a combination of theoretical explanations, examples, proofs, and applications in abstract algebra. The course also includes practical demonstrations using tools like Plotly in a Google Colaboratory notebook.
The intended audience for this course includes students and learners interested in advanced mathematics, specifically abstract algebra. This course is suitable for individuals with a basic understanding of algebra and mathematical concepts.
Overview
Syllabus
01 Introduction.
02 Introduction to sets.
03 Subsets.
04 Our first proof.
05 Products sets and mappings.
06 More about mappings.
08 Exercise problem.
09 Exercise problem.
10 Relations (still with the not-so-exciting-stuff).
11 Reflexive, symmetric, and transitive properties of relations.
12 Equivalence relations.
13 Equivalence sets.
14 Ordering of sets.
15 Properties of partially ordered sets.
16 You have made it to the first exciting video Operations.
17 Examples of binary operations.
18 Types of binary operations.
19 Defining the types of binary operations.
20 The identity element.
21 The inverse element.
22 Combinations of binary operations.
23 Algebraic system isomorphism.
Definition of a group Lesson 24.
Cancellation law Lesson 25.
Group equation examples Lesson 26.
Groups that commute Lesson 27.
Example problems with inverses Lesson 28.
Integers modulo n.
Groups in abstract algebra examples.
Subgroups abstract algebra.
Cyclic groups and generators.
Symmetric groups.
Homomorphisms in abstract algebra.
Homomorphisms in abstract algebra examples.
Isomorphisms in abstract algebra.
Proof that Cayley table row and column entries are unique and complete.
Cayley theorem proof.
Cosets in abstract algebra.
Cosets and equivalence class proof.
Coset example.
Coset deeper insights.
Normal subgroups.
Cosets generated by elements of cosets.
Lagrange theorem.
Quotient groups.
Quotient group example.
Product groups.
Product group example.
Surjective homomorphisms in abstract algebra.
Kernel of a group homomorphism.
First theorem of isomorphisms.
Group actions in abstract algebra.
Group action proofs in abstract algebra.
Group action examples.
Orbit of a set in abstract algebra.
Stabilizer in abstract algebra.
Example of a stabilizer in abstract algebra.
Orbit stabilizer theorem.
Dihedral groups in abstract algebra.
Dihedral group example.
Center of a group in abstract algebra.
Centralizer of an element in the dihedral group of order 6.
Centralizer of a set in a group.
Normalizer of a set in a group.
Conjugation of a group.
Conjugacy classes of a group.
Class equation of a group.
Cycle types and conjugacy classes of the symmetric group.
Group automorphisms in abstract algebra.
Group automorphism example.
How to use Plotly in a Google Colaboratory notebook.
Taught by
Dr Juan Klopper
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Reviews
5.0 rating, based on 2 Class Central reviews
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This was a great way to introduce myself to Abstract Algebra. I greatly appreciated the examples to make sure I understood the material.
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Md Jalal MollaI am so proud to have done the course successfully. I have known many many thins about Abstract Algebra. This course was helpful for me and I have benefited from the course.
