I Group Labs.- 1 Using Symmetry to Uncover a Group.- Getting started? Begin here.- A symmetry of an equilateral triangle.- Are there other symmetries?.- Multiplying the transformations.- Are there any commuters?.- Is it always bad to be closed-minded?.- We should try to find our identity.- Is it perverse to not have an inverse?.- Should we associate together?.- What else?.- Let's group it all together.- 2 Determining the Symmetry Group of a Given Figure.- Symmetries and how to find them.- Your turn.- 3 Is This a Group?.- When do we have a group?.- Your turn.- 4 Let's Get These Orders Straight.- Order of g and its inverse.- Distribution of the orders of elements in ?n.- Another look at orders.- What is P( \ g.- More questions about Un.- 5 Subversively Grouping Our Elements.- When do we have a subgroup?.- Subgroups of ?n.- P(H < G) for a random subset H ofG = ?n.- Necessary elements for full closure.- Subgroups of Un.- 6 Cycling Through the Groups.- What, when, how, and why about cyclic groups.- Cyclicity of ?m ? ?n.- Structure of intersections of subgroups of ?.- 7 Permutations.- What is a permutation?.- Computations with permutations.- Applications of permutations.- Questions about permutations.- 8 Isomorphisms.- What is an isomorphism?.- Creating Morphoids.- Seeing isomorphisms.- 9 Automorphisms.- Automorphisms on ?n.- Inner automorphisms.- 10 Direct Products.- What is a direct product?.- Order of an element in a direct product.- When is a direct product of cyclic groups cyclic?.- Isomorphisms among Un groups.- 11 Cosets.- Cosets, left and right.- Properties of cosets.- 12 Normality and Factor Groups.- Normal subgroups.- Making a new group.- Factor groups.- 13 Group Homomorphisms.- What is a group homomorphism?.- The kernel and image.- Properties that are preserved by homomorphisms.- The kernel is normal.- The First Homomorphism Theorem.- The alternating group-parity as a morphism.- 14 Rotational Groups of Regular Polyhedra.- The rotational group of the tetrahedron.- Further exercises.- II Ring Labs.- 1 Introduction to Rings and Ringoids.- Getting started? Begin here.- Ringoids and rings.- Properties of rings.- Additional exercises.- 2 Introduction to Rings, Part 2.- Units and zero divisors.- Integral domains.- Fields.- Additional exercises.- 3 An Ideal Part of Rings.- What is the ideal part of a ring?.- Ideals factor into other ring properties.- 4 What Does ?[i](a + b i) Look Like?.- Example 1.- Example 2.- 5 Ring Homomorphisms.- Morphoids on rings.- Ring homomorphisms.- The kernel and image.- The kernel is an ideal.- One-rule Morphoids.- The Chinese Remainder Theorem.- 6 Polynomial Rings.- to polynomials.- Divide and conquer.- 7 Factoring and Irreducibility.- to factoring and irreducibility.- Some techniques on testing the irreducibility of polynomials.- More polynomials for practice.- Toolbox of theorems.- Final perspective.- 8 Roots of Unity.- A closer look-graphically.- Another look-algebraically.- 9 Cyclotomic Polynomials.- Search for gn(x).- Some properties of ?x(x).- 10 Quotient Rings of Polynomials.- Polynomials over a field.- A homomorphism based on PolynomialRemainder.- Defining a quotient ring of polynomials.- The PolynomialRemainder function ? is indeed a homomorphism.- Is V a field?.- Is V what we claimed?.- 11 Quadratic Field Extensions.- The general problem.- An extension of ?3 using Mathematica.- Theorems motivated from this lab.- 12 Factoring in ?[?d].- to divisibility.- Associates, irreducibility, and norms.- Units in?[?d].- Factoring 46 in ?[?5].- Is ?[?6] a UFD?.- 13 Finite Fields.- Creation of finite fields.- Finite field theorems and illustrations.- III User's Guide.- 1 Introduction to Abstract Algebra.- Packages in AbstractAlgebra.- Basic structures used in AbstractAlgebra.- How to use a Mode.- Using Visual mode with "large" elements.- How to change the Structure.- 2 Groupolds.- Forming Groupoids.- Structure of Groupoids.- Testing the defining properties of a group.- Built-in groupoids.- Uses of the Cayley table.- Building other structures.- Other group properties.- 3 Ringoids.- Forming Ringoids.- Structure of Ringoids.- Testing properties of a ring.- Built-in Ringoids.- Using Cayley tables.- Building other structures.- Extension ringoids.- Polynomials over a ringoid.- Matrices over a ringoid.- Functions on a ringoid.- Finite fields.- 4 Morphoids.- Forming Morphoids.- Structure of Morphoids.- Built-in Morphoids.- Properties.- Kernel, Image, and InverseImage.- Automorphisms.- Visualizing Morphoids.- 5 Additional Functionality.- Global variables and options.- Working with permutations and cycles.- Working in ?[?d].- Miscellaneous functions.
