Abstract Algebra: Introduction To Group Theory

Group Theory

What you’ll learn
Basic notions in Set Theory including relations, equivalence relations, partitions of sets
maps, and their compositions, permutations
Binary operations, mathematical systems, the definition of a group, properties of groups
Subgroups, generating subgroups, and normal subgroups
Cosets of subgroups, Lagrange’s Theorem, Sylow Theorems, Quotient groups
Homomorphisms, isomorphisms, and automorphisms of groups
Requirements
Basic training in mathematics
Description
Group theory is a branch of abstract algebra that studies the algebraic structures known as groups. This theory is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and modules, can all be seen as groups endowed with additional operations and axioms. Many physical systems, such as crystals, can be modeled by symmetric groups. Group theory has many vital applications in physics, chemistry, and materials science. Group theory is also central to public-key cryptography. Group Theory has applications in other areas of mathematics such as Galois Theory where it is shown that the result that S_5, the symmetric group in five elements, is not solvable, implies that the general quintic equation cannot be solved by radicals in the way equations of lower degree can.This course is to give an introduction to Group Theory. It covers the following topics:1. Basic notions in Set Theory including relations, equivalence relations, partitions of sets, maps, and their compositions, permutations2. Binary operations, mathematical systems, the definition of a group, properties of groups3. Subgroups, generating subgroups, and normal subgroups4. Cosets of subgroups, Lagrange’s Theorem, Sylow Theorems, Quotient groups5. Homomorphisms, isomorphisms, and automorphisms of groupsThe course is taught in a way that can be understood easily.

Overview

Section 1: Basics

Lecture 1 Relations

Lecture 2 Equivalence Relations

Lecture 3 Partitions of Sets

Lecture 4 Maps

Lecture 5 The Composition of Maps

Lecture 6 Invertible Maps

Lecture 7 Permutations

Section 2: Groups

Lecture 8 Binary Operations

Lecture 9 Mathematical Systems

Lecture 10 The Definition of a Group

Lecture 11 Some Properties of Groups

Lecture 12 Equivalent Definitions of a Group

Section 3: Subgroups

Lecture 13 Definition

Lecture 14 Generating subgroups

Lecture 15 Cosets of subgroups

Lecture 16 Lagrange’s theorem

Lecture 17 The Sylow theorems

Section 4: Normal Subgroups

Lecture 18 Definition

Lecture 19 Products of subgroups

Section 5: Quotient Groups

Lecture 20 Quotient groups

Section 6: Isomorphism Theorems

Lecture 21 Homomorphisms of groups

Lecture 22 Isomorphisms and Automorphisms

Lecture 23 The isomorphism theorem

Undergraduate students