This course introduces abstract algebra, also known as modern algebra. It will cover topics such as group definition and motivation, multiplication tables and Cayley tables, cosets and Lagrange’s Theorem, normal subgroups and quotient groups, cyclic groups and group homomorphisms, isomorphisms and the kernel of a group homomorphism, the order of an element, symmetric groups and cycle notation of permutations, dihedral groups and symmetric groups of triangles, matrix groups and direct products of groups, simple groups, ring definition and examples, units in a ring, integral domains, ideals in ring theory, field definition and examples, vector spaces and modules. It also draws connections between abstract algebra and other fields of math, such as geometry, number theory and linear algebra. This course is designed to lay a foundation of knowledge to help students understand abstract algebra and its applications.
Overview
Syllabus
What is Abstract Algebra? (Modern Algebra).
Group Definition (expanded) - Abstract Algebra.
Abstract Algebra: The definition of a Group.
Abstract Algebra: Motivation for the definition of a group.
Abstract Algebra: The definition of a Subgroup.
Group Multiplication Tables | Cayley Tables (Abstract Algebra).
Cosets and Lagrange’s Theorem - The Size of Subgroups (Abstract Algebra).
Normal Subgroups and Quotient Groups (aka Factor Groups) - Abstract Algebra.
Cyclic Groups (Abstract Algebra).
Abstract Algebra: Group or Not Group? (Integer edition).
Group Homomorphisms - Abstract Algebra.
Homomorphisms (Abstract Algebra).
Isomorphisms (Abstract Algebra).
The Kernel of a Group Homomorphism – Abstract Algebra.
The Order of an Element (Abstract Algebra).
Symmetric Groups (Abstract Algebra).
Cycle Notation of Permutations - Abstract Algebra.
Dihedral Group (Abstract Algebra).
Symmetry Groups of Triangles (Abstract Algebra).
Matrix Groups (Abstract Algebra).
Direct Products of Groups (Abstract Algebra).
Simple Groups - Abstract Algebra.
Abstract Algebra: The definition of a Ring.
Ring Definition (expanded) - Abstract Algebra.
Ring Examples (Abstract Algebra).
Units in a Ring (Abstract Algebra).
Integral Domains (Abstract Algebra).
Ideals in Ring Theory (Abstract Algebra).
Field Definition (expanded) - Abstract Algebra.
Abstract Algebra: The definition of a Field.
Field Examples - Infinite Fields (Abstract Algebra).
What is a Vector Space? (Abstract Algebra).
What is a Module? (Abstract Algebra).
Taught by
Socratica
