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

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