Exploring Abstract Algebra with Mathematica

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This subsection gives an expanded Table of Contents for Exploring Abstract Algebra with Mathematica. Either scroll through the material or use a link below to jump to the desired portion. For an abbreviated version of the TOC, see the Table of Contents submenu on the right. (If desired, the essence of what is given below is also available in the form of a Mathematica notebook.)

• Group Lab 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14
• Ring Lab 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13
• User's Guide Introduction to AbstractAlgebra, Groupoids, Ringoids, Morphoids, Additional Functionality
• Appendix A, B

Preface

Part I -- Group Labs

• Lab 1. Using Symmetry to Uncover a Group
  • 1.0 Note regarding Exploring Abstract Algebra with Mathematica
  • 1.1 Prerequisites
  • 1.2 Goals for this lab
  • 1.3 Getting started? - begin here
  • 1.4 A symmetry of an equilateral triangle
  • 1.5 Are there other symmetries?
  • 1.6 Multiplying our transformations
  • 1.7 Are there any commuters?
  • 1.8 Is it always bad to be closed-minded?
  • 1.9 We should try to find our identity
  • 1.10 Is it perverse to not have an inverse?
  • 1.11 Should we associate together?
  • 1.12 What else?
  • 1.13 Let's group it all together
  • 1.14 Mathematica commands used in this lab
• Lab 2. Determining the Symmetry Group of a Given Figure
  • 2.0 Note regarding Exploring Abstract Algebra with Mathematica
  • 2.1 Prerequisites
  • 2.2 Goals for this lab
  • 2.3 Symmetries and how to find them
  • 2.4 Now your turn
  • 2.5 Mathematica commands used in this lab
• Lab 3. Is This a Group?
  • 3.0 Note regarding Exploring Abstract Algebra with Mathematica
  • 3.1 Prerequisites
  • 3.2 Goals for this lab
  • 3.3 When do we have a group?
  • 3.4 Now your turn
  • 3.5 Mathematica commands used in this lab
• Lab 4. Let's Get These Orders Straight
  • 4.0 Note regarding Exploring Abstract Algebra with Mathematica
  • 4.1 Prerequisites
  • 4.2 Goals for this lab
  • 4.3 Order of g and its inverse
  • 4.4 Distribution of the orders of elements in
  • 4.5 Another look at orders
  • 4.6 What is for g in ?
  • 4.7 More questions about
  • 4.8 Mathematica commands used in this lab
• Lab 5. Subversively Grouping Our Elements
  • 5.0 Note regarding Exploring Abstract Algebra with Mathematica
  • 5.1 Prerequisites
  • 5.2 Goals for this lab
  • 5.3 When do we have a subgroup?
  • 5.4 Subgroups of
  • 5.5 for a random subset H of
  • 5.6 Necessary elements for full closure
  • 5.7 Subgroups of
  • 5.8 Mathematica commands used in this lab
• Lab 6. Cycling Through the Groups
  • 6.0 Note regarding Exploring Abstract Algebra with Mathematica
  • 6.1 Prerequisites
  • 6.2 Goals for this lab
  • 6.3 What, when, how and why about cyclic groups
  • 6.4 Cyclicity of
  • 6.5 Structure of intersections of subgroups of Z
  • 6.6 Mathematica commands used in this lab
• Lab 7. Permutations
  • 7.0 Note regarding Exploring Abstract Algebra with Mathematica
  • 7.1 Prerequisites
  • 7.2 Goals for this lab
  • 7.3 What is a permutation?
  • 7.4 Computations with permutations
  • 7.5 Applications of permutations
  • 7.6 Questions about permutations
  • 7.7 Mathematica commands used in this lab
• Lab 8. Isomorphisms
  • 8.0 Note regarding Exploring Abstract Algebra with Mathematica
  • 8.1 Prerequisites
  • 8.2 Goals for this lab
  • 8.3 What is an isomorphism?
  • 8.4 Creating Morphoids
  • 8.5 Seeing isomorphisms
  • 8.6 Mathematica commands used in this lab
• Lab 9. Automorphisms
  • 9.0 Note regarding Exploring Abstract Algebra with Mathematica
  • 9.1 Prerequisites
  • 9.2 Goals for this lab
  • 9.3 Automorphisms on
  • 9.4 Inner automorphisms
  • 9.5 Mathematica commands used in this lab
• Lab 10. Direct Products
  • 10.0 Note regarding Exploring Abstract Algebra with Mathematica
  • 10.1 Prerequisites
  • 10.2 Goals for this lab
  • 10.3 What is a direct product?
  • 10.4 Order of an element in a direct product
  • 10.5 When is a direct product of cyclic groups cyclic?
  • 10.6 Isomorphisms among groups
  • 10.7 Mathematica commands used in this lab
• Lab 11. Cosets
  • 11.0 Note regarding Exploring Abstract Algebra with Mathematica
  • 11.1 Prerequisites
  • 11.2 Goals for this lab
  • 11.3 Cosets, left and right
  • 11.4 Properties of cosets
  • 11.5 Mathematica commands used in this lab
• Lab 12. Normality and Factor Groups
  • 12.0 Note regarding Exploring Abstract Algebra with Mathematica
  • 12.1 Prerequisites
  • 12.2 Goals for this lab
  • 12.3 Normal subgroups
  • 12.4 Making a new group
  • 12.5 Factor groups
  • 12.6 Mathematica commands used in this lab
• Lab 13. Group Homomorphisms
  • 13.0 Note regarding Exploring Abstract Algebra with Mathematica
  • 13.1 Prerequisites
  • 13.2 Goals for this lab
  • 13.3 What is a group homomorphism?
  • 13.4 The kernel and image
  • 13.5 Properties that are preserved by homomorphisms
  • 13.6 The kernel is normal
  • 13.7 The First Homomorphism Theorem
  • 13.8 The alternating group -- parity as a morphism
  • 13.9 Mathematica commands used in this lab
• Lab 14: Rotational Groups of Regular Polyhedra
  • 14.0 Note regarding Exploring Abstract Algebra with Mathematica
  • 14.1 Prerequisites
  • 14.2 Goals for this lab
  • 14.3 Example - the Tetrahedron
  • 14.4 Further exercises
  • 14.5 Mathematica commands used in this lab

Part II -- Ring Labs

• Lab 1. Introduction to Rings and Ringoids
  • 1.0 Note regarding Exploring Abstract Algebra with Mathematica
  • 1.1 Prerequisites
  • 1.2 Goals for this lab
  • 1.3 Getting started? Begin here
  • 1.4 Ringoids and rings
  • 1.5 Properties of rings
  • 1.6 Additional exercises
  • 1.7 Mathematica commands used in this lab
• Lab 2. Introduction to Rings, Part 2
  • 2.0 Note regarding Exploring Abstract Algebra with Mathematica
  • 2.1 Prerequisites
  • 2.2 Goals for this lab
  • 2.3 Units & zero divisors
  • 2.4 Integral domains
  • 2.5 Fields
  • 2.6 Additional exercises
  • 2.7 Mathematica commands used in this lab
• Lab 3. An Ideal Part of Rings
  • 3.0 Note regarding Exploring Abstract Algebra with Mathematica
  • 3.1 Prerequisites
  • 3.2 Goals for this lab
  • 3.3 What is the ideal part of a ring?
  • 3.4 Ideals factor into other ring properties
  • 3.5 Mathematica commands used in this lab
• Lab 4. What Does Look Like?
  • 4.0 Note regarding Exploring Abstract Algebra with Mathematica
  • 4.1 Prerequisites
  • 4.2 Goals for this lab
  • 4.3 First example
  • 4.4 A second example
  • 4.5 Mathematica commands used in this lab
• Lab 5. Ring Homomorphisms
  • 5.0 Note regarding Exploring Abstract Algebra with Mathematica
  • 5.1 Prerequisites
  • 5.2 Goals for this lab
  • 5.3 Morphoids on rings
  • 5.4 Ring homomorphisms
  • 5.5 The kernel and image
  • 5.6 The kernel is an ideal
  • 5.7 One rule Morphoids
  • 5.8 The Chinese Remainder Theorem
  • 5.9 Mathematica commands used in this lab
• Lab 6. Polynomial Rings
  • 6.0 Note regarding Exploring Abstract Algebra with Mathematica
  • 6.1 Prerequisites
  • 6.2 Goals for this lab
  • 6.3 An introduction to polynomials
  • 6.4 Divide and conquer
  • 6.5 Mathematica commands used in this lab
• Lab 7. Factoring and Irreducibility
  • 7.0 Note regarding Exploring Abstract Algebra with Mathematica
  • 7.1 Prerequisites
  • 7.2 Goals for this lab
  • 7.3 An introduction to factoring and irreducibility
  • 7.4 Some techniques on testing the irreducibility of polynomials
  • 7.5 More polynomials for practice
  • 7.6 Toolbox of theorems
  • 7.7 A final perspective
  • 7.8 Mathematica commands used in this lab
• Lab 8. Roots of Unity
  • 8.0 Note regarding Exploring Abstract Algebra with Mathematica
  • 8.1 Prerequisites
  • 8.2 Goals for this lab
  • 8.3 An introduction
  • 8.4 A closer look -- graphically
  • 8.5 Another look -- algebraically
  • 8.6 Mathematica commands used in this lab
• Lab 9. Cyclotomic Polynomials
  • 9.0 Note regarding Exploring Abstract Algebra with Mathematica
  • 9.1 Prerequisites
  • 9.2 Goals for this lab
  • 9.3 An introduction
  • 9.4 The search for
  • 9.5 Some properties of
  • 9.6 Mathematica commands used in this lab
• Lab 10. Quotient Rings of Polynomials
  • 10.0 Note regarding Exploring Abstract Algebra with Mathematica
  • 10.1 Prerequisites
  • 10.2 Goals for this lab
  • 10.3 Polynomials over a field
  • 10.4 A homomorphism based on PolynomialRemainder
  • 10.5 Defining a quotient ring of polynomials
  • 10.6 The PolynomialRemainder function theta is indeed a homomorphism
  • 10.7 Is V a field?
  • 10.8 Is V what we claimed?
  • 10.9 Mathematica commands used in this lab
• Lab 11. Quadratic Field Extensions
  • 11.0 Note regarding Exploring Abstract Algebra with Mathematica
  • 11.1 Prerequisites
  • 11.2 Goals for this lab
  • 11.3 The general problem
  • 11.4 An extension of using Mathematica
  • 11.5 Theorems that are motivated from this lab
  • 11.6 Mathematica commands used in this lab
• Lab 12. Factoring in
  • 12.0 Note regarding Exploring Abstract Algebra with Mathematica
  • 12.1 Prerequisites
  • 12.2 Goals for this lab
  • 12.3 An introduction to divisibility
  • 12.4 Associates, irreducibility and norms
  • 12.5 Units in
  • 12.6 Factoring 46 in
  • 12.7 Is a UFD?
  • 12.8 Mathematica commands used in this lab
• Lab 13. Finite Fields
  • 13.0 Note regarding Exploring Abstract Algebra with Mathematica
  • 13.1 Prerequisites
  • 13.2 Goals for this lab
  • 13.3 Creation of finite fields
  • 13.4 Finite field theorems and illustrations
  • 13.5 Mathematica commands used in this lab

Part III -- User's Guide

• Introduction to AbstractAlgebra
  • 1.0 Read me first
  • 1.1 Packages in AbstractAlgebra
  • 1.2 Basic structures used in AbstractAlgebra
    • 1.2.1 Overview
    • 1.2.2 How to form Groupoids, Ringoids and Morphoids
  • 1.3 How to use a Mode
  • 1.4 Using Visual mode with "large" elements
  • 1.5 How to change the Structure
• Groupoids
  • 2.0 Read me first
  • 2.1 Forming Groupoids
    • 2.1.1 FormGroupoid
    • 2.1.2 GenerateGroupoid
    • 2.1.3 FormGroupoidByTable
    • 2.1.4 FormGroupoidFromCycles and RandomGroupoid
  • 2.2 The structure of Groupoids
  • 2.3 Testing the defining properties of a group
    • 2.3.1 The four standard functions
    • 2.3.2 Related functions
  • 2.4 Built-in groupoids
    • 2.4.1 Groupoids based on the integers mod
    • 2.4.2 Other numeric-based groupoids
    • 2.4.3 Groups of permutations
    • 2.4.4 Dihedral and Cyclic groups
    • 2.4.5 Other groupoids
  • 2.5 Uses of the Cayley table
  • 2.6 Building other structures
    • 2.6.1 Direct products
    • 2.6.2 Subgroups
    • 2.6.3 Quotient groups
  • 2.7 Other group properties
• Ringoids
  • 3.0 Read me first
  • 3.1 Forming Ringoids
    • 3.1.1 FormRingoid
    • 3.1.2 FormRingoidByTable
  • 3.2 The Structure of Ringoids
    • 3.2.1 Basic functions
    • 3.2.2 Related functions
    • 3.2.3 Groupoids from ringoids
  • 3.3 Testing properties of a ring
    • 3.3.1 Additive properties
    • 3.3.2 Multiplicative properties
    • 3.3.3 The distributive property
    • 3.3.4 RingQ test
    • 3.3.5 Specialized rings
    • 3.3.6 Closure of subsets
    • 3.3.7 Testing other properties
  • 3.4 Built-in Ringoids
    • 3.4.1 Numeric Rings
    • 3.4.2 Other Rings
  • 3.5 Using Cayley tables
  • 3.6 Building other structures
    • 3.6.1 Direct Products
    • 3.6.2 Subrings and ideals
    • 3.6.3 Quotient Rings
  • 3.7 Extension ringoids
  • 3.8 Polynomials over a ringoid
    • 3.8.1 Forming polynomials
    • 3.8.2 Random polynomials
    • 3.8.3 Polynomial arithmetic
    • 3.8.4 Quotient rings of polynomials
    • 3.8.5 Irreducibility of integer-based polynomials
    • 3.8.6 Functions related to solving equations or evaluation
    • 3.8.7 Extensions of ordinary Mathematica functions
    • 3.8.8 Miscellaneous functions
  • 3.9 Matrices over a ringoid
    • 3.9.1 Individual matrices
    • 3.9.2 Matrix arithmetic
    • 3.9.3 Determinants & inverses
    • 3.9.4 Matrix ringoids
    • 3.9.5 Matrix groupoids
    • 3.9.6 Miscellaneous functions
  • 3.10 Functions on a ringoid
    • 3.10.1 Function extensions and their elements
    • 3.10.2 Function arithmetic
    • 3.10.3 Polynomial conversion and interpolation
  • 3.11 Finite fields
• Morphoids
  • 4.0 Read me first
  • 4.1 Forming Morphoids
  • 4.2 The structure of Morphoids
  • 4.3 Built-in Morphoids
  • 4.4 Properties
    • 4.4.1 surjectivity and injectivity
    • 4.4.2 preserving operations
  • 4.5 Kernel, Image, and InverseImage(s)
  • 4.6 Automorphisms
  • 4.7 Visualizing Morphoids
• Additional Functionality
  • 5.0 Read me first
  • 5.1 Global variables and options
  • 5.2 Working with permutations and cycles
    • 5.2.1 Introduction
    • 5.2.2 Permutation operations
    • 5.2.3 Representing permutations
    • 5.2.4 Cycles
    • 5.2.5 Cycle operations
    • 5.2.6 Other cycle-related functions
    • 5.2.7 Stabilizers and orbits
  • 5.3 Working in
    • 5.3.1 Basic functions
    • 5.3.2 Divisibility
    • 5.3.3 Norm-related functions
  • 5.4 Miscellaneous functions
    • 5.4.1 Working with lists
    • 5.4.2 Working with graphics
    • 5.4.3 Adjoin
    • 5.4.4 Disguising groups and rings
    • 5.4.5 A look at some functions in LabCode
    • 5.4.6 Potpourri

Appendices

• Appendix A -- Installation Instructions and References
  • 6.1 Installation instructions
    • 6.1.1 Version 3.0 or higher
    • 6.1.2 Version 2.x under Windows
    • 6.1.3 Version 2.x under other platforms
  • 6.2 Version-specific notes
    • 6.2.1 Version 3.0 or higher
    • 6.2.2 Version 2.x under Windows
    • 6.2.3 Version 2.x under other platforms
  • 6.3 References
    • 6.3.1 Mathematica references
    • 6.3.2 Abstract Algebra references
    • 6.3.3 Other software tools for Abstract Algebra
  • 6.4 Objects in AbstractAlgebra
    • 6.4.1 Alphabetically
    • 6.4.2 By Packages
    • 6.4.3 Objects extended from standard Mathematica
  • 6.5 Description of Exploring Abstract Algebra with Mathematica
    • 6.5.1 Overview
    • 6.5.2 Group labs
    • 6.5.3 Ring labs
• Appendix B -- Lab 0 Getting Started with Mathematica
  • 0.0 Note regarding Exploring Abstract Algebra with Mathematica
  • 0.1 Prerequisites
  • 0.2 Goals for this lab
  • 0.3 The In's and Out's of evaluating Mathematica expressions
  • 0.4 What is inside?
  • 0.5 Some syntax basics
  • 0.6 Help
  • 0.7 Using Mathematica to learn a mod idea
  • 0.8 Divide and conquer
  • 0.9 It all adds up
  • 0.10 Mathematica commands used in this lab

Index

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Prepared by Al Hibbard. Most recent update: 6/16/2006. This page has been viewed 277 times since June 16, 2006. The entire EAAM web site has had 669,335 hits since July 23, 2002.