Doc Benton's Academy for Mathematical Wizards!


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Welcome to "Doc Benton's Academy for Mathematical Wizards!" I am recently retired, and this site will be under reconstruction for at least the coming the year. Nonetheless, below is what I have accomplished so far in my quest to create materials on group theory that an intelligent high school student could benefit from. Right now I am just getting the bare bones in place, and when I'm done with that, I will clean up and simplify the text as much as possible in order to try and make everything perfectly clear and unambiguous. Enjoy!

Beginning Group Theory

This first section teaches you a lot about group theory without delving into proofs. Many times, instead of a formal proof, I just give a typical example. Furthermore, each lesson is followed by some practice problems and answers, and then this is quite often followed by a related Youtube video. The videos generally explain or do things differently or introduce additional notation, and so paying attention to them will give your mind increased fexibility in dealing with groups. Groups are wonderful!

What Is a Group? What Is a Group - Practice What Is a Group - Answers Introduction to Group Theory (video) Introduction to Cycle Notation (video) How to Multiply Cycles Left to Right (video) Group Theory 101 (video) Cyclic Groups and Subgroups Cyclic Groups and Subgroups - Practice Cyclic Groups and Subgroups - Answers
Modular Arithmetic (video) Definition of a Cyclic Group (video) Cyclic Subgroups (video) Dihedral Groups Dihedral Groups - Practice Dihedral Groups - Answers Dihedral Groups (video) Symmetric Groups Symmetric Groups - Practice Symmetric Groups - Answers
Symmetric Groups (video) Direct Products Direct Products - Practice Direct Products - Answers Direct Products of Groups (video) Direct Products of Finite Cyclic Groups (video) The Klein 4-group (video) Generator Diagrams Generator Diagrams - Practice Generator Diagrams - Answers
Cycle Graphs Cycle Graphs - Practice Cycle Graphs - Answers Cayley Graphs Cayley Graphs - Practice Cayley Graphs - Answers Subgroup Lattices Subgroup Lattices - Practice Subgroup Lattices - Answers Introduction to Conjugates
Introduction to Conjugates - Practice Introduction to Conjugates - Answers Introduction to Commutators Introduction to Commutators - Practice Introduction to Commutators - Answers Commutators - Cube theory 101 (video) Rubik's Cube Theory: Commutators (video) Rubik's - Build Your own Algortihms with Commutators (video) Introduction to Orbits Introduction to Orbits - Practice
Introduction to Orbits - Answers Orbits and Cycles (video) Introduction to the Stabilizer Subgroup Introduction to the Stabilizer Subgroup - Practice Introduction to the Stabilizer Subgroup - Answers Stabilizer of Group Action on a Set (video) Equivalence Relations Equivalence Relations - Practice Equivalence Relations - Answers Equivalence Relations (video)
Equivalence Relations (a different video) Definition of an Equivalence Relation (video) Equivalence Classes (video) Cosets Cosets - Practice Cosets - Answers Cosets of a Subgroup (video) Definition of Cosets (video) Cosets and Lagrange's Theorem (video) Left Cosets and Right Cosets are the Same Size (video)
Cosets are Equivalence Classes (video) Why normal subgroups are so important Why normal subgroups are so important - Practice Why normal subgroups are so important - Answers Quotient Group of a Normal Subgroup (video) More on Quotient Groups More on Quotient Groups - Practice More on Quotient Groups - Answers The Fundamental Theorem of Finite Abelian Groups The Fundamental Theorem of Finite Abelian Groups - Practice
The Fundamental Theorem of Finite Abelian Groups - Answers The Fundamental Theorem of Finite Abelian Groups (video) Semidirect Products Semidirect Products - Practice Semidirect Products - Answers Frieze Groups Frieze Groups - Practice Frieze Groups - Answers Frieze Patterns in Maths and Architecture (video) Frieze Patterns (video)
The Seven Geese Groups (video) Wallpaper Groups Wallpaper Groups - Practice Wallpaper Groups - Answers How toCreate a Tesselations (video) Escher's Tesselations (video) Free Range Symmetry Free Range Symmetry - Practice Group Theory as an Act of Creation Group Theory as an Act of Creation - Practice
Installing GAP Software How to Use GAP Abstract Algebra with GAP (manual) Small Groups Klein 4-Group Dihedral Group of Degree 3 Quaternion Group Z3 Semidirect Product Z4 Alternating Group of Degree 4 Cyclic Group of Order 12


Intermediate Group Theory

Introduction to Intermediate Group Theory Symbolic Logic - Practice Symbolic Logic - Answers Set Theory - Practice Set Theory - Answers Set Theory Continued - Practice Set Theory Continued - Answers Mathematical Induction - Practice Mathematical Induction - Answers Everything Else I Forgot to Mention - Practice
Everything Else I Forgot to Mention - Answers Index of Theorems Theorem 1 - Practice Theorem 1 - Answer Theorem 2 - Practice Theorem 2 - Answer Theorem 3 - Practice Theorem 3 - Answer Theorem 4 - Practice Theorem 4 - Answer
Theorem 5 - Practice Theorem 5 - Answer Theorem 6 - Practice Theorem 6 - Answer Theorem 7 - Practice Theorem 7 - Answer Theorem 8 - Practice Theorem 8 - Answer Theorem 9 - Practice Theorem 9 - Answer
Theorem 10 - Practice Theorem 10 - Answer Theorem 11 - Practice Theorem 11 - Answer Theorem 12 - Practice Theorem 12 - Answer Theorem 13 - Practice Theorem 13 - Answer Theorem 14 - Practice Theorem 14 - Answer
Theorem 15 - Practice Theorem 15 - Answer Theorem 16 - Practice Theorem 16 - Answer Theorem 17 - Practice Theorem 17 - Answer Theorem 18 - Practice Theorem 18 - Answer Theorem 19 - Practice Theorem 19 - Answer
Theorem 20 - Practice  Theorem 20 - Answer Theorem 21 - Practice Theorem 21 - Answer Theorem 22 - Practice Theorem 22 - Answer Theorem 23 - Practice Theorem 23 - Answer Theorem 24 - Practice Theorem 24 - Answer
Theorem 25 - Practice Theorem 25 - Answer Theorem 26 - Practice Theorem 26 - Answer Theorem 27 - Practice Theorem 27 - Answer Theorem 28 - Practice Theorem 28 - Answer Theorem 29 - Practice Theorem 29 - Answer
Theorem 30 - Practice Theorem 30 - Answer Theorem 31 - Practice Theorem 31 - Answer Theorem 32 - Practice Theorem 32 - Answer Theorem 33 - Practice Theorem 33 - Answer Theorem 34 - Practice Theorem 34 - Answer
Theorem 35 - Practice Theorem 35 - Answer Theorem 36 - Practice Theorem 36 - Answer Theorem 37 - Practice Theorem 37 - Answer Theorem 38 - Practice Theorem 38 - Answer Theorem 39 - Practice Theorem 39 - Answer


Advanced Group Theory

Isomorphisms and Homomorphisms The Kernel of a Homomorphism The Natural Homomorphism The Correspondence Theorem Homomorphisms and Identities Homomorphisms and Inverses Homomorphism and One-To-One Functions The First Isomorphism Theorem The Second Isomorphism Theorem The Third Isomorphism Theorem
An Application Another Application A Third Application A Fourth Result Quotients of Quotients of Quotients Orbits Orbits-Exercises Orbits-Exercises-Answers Conjugates The Sylow Theorems
The Fundamental Theorem of Finite Abelian Groups - Proof Greatest Common Divisors                


Rubik's Cube

Introduction to Rubik's Cube Rubik Software Rubik's Cube Solution Permutations Practice -Permutations Practice -Permutations -Answers Counting the Number of Permutations in Rubik's Cube What is a Group? Practice -What is a Group? Practice -What is a Group -Answers
Some Special Classes of Groups Practice -Some Special Classes of Groups Practice -Some Special Classes of Groups -Answers Subgroups Practice -Subgroups Practice -Subgroups -Answers Rubik's Cube Subgroups Conjugates and Commutators Practice -Conjugates and Commutators Practice -Conjugates and Commutators Answers
Conjugates and Commutators in Rubik's Cube Patterns on Rubik's Cube Revisiting the Solution to Rubik's Cube Normal Subgroups and Homomorphisms Practice -Normal Subgroups and Homomorphisms Practice -Normal Subgroups and Homomorphisms -Answers Visual Group Theory Practice -Visual Group Theory Practice- Visual Group Theory -Answers Groups of Small Order
Life Lessons from Group Theory and Rubik's Cube Group Explorer                

General Math

Statistics Business Calculus Advanced Business Math Multivariable Calculus Pi Day! The Theology of Pi Infinity Einstein's Theory of Special Relativity E=mc-squared in One Page! Final Talk
Reflections on Educational Practices in
Rabbbinic Literature
Best Practices Algebraic Models for Constructivist Theories of Perception Factoring Examples How to factor trinomials - proof How to Solve a Cubic Equation How to Solve a Quartic Equation Doc Benton's Video Ranch Math Song  


Statistics Examples

Statistics and Austerity Economics

Statistics and Gun Laws

Statistics and the Media

Statistics and Biased Studies

Statistics and Reporting

Statistics and Measurement Error

Statistics and Causation

Statistics and Climate Change

Statistics and Graphs

Statistics and Demographic Changes

Statistics and Early Death

Statistics and Biased Interpretations

Statistics and Biased Samples

Statistics and Not So Good Graphs

Sampling Error


Differential Equations

What is a Differential Equation? Separable Differential Equations Exact Differential Equations Integrating Factors Constant Coefficients Springs The Incredible Harmonic Oscillator! Laplace Transforms Doc Benton's Favorite Laplace Transforms Gamma Function