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All Problems listed in Problem Order
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Alphabetical List of Problems
- Collapsing A Factor Of An External Direct Product Yields The Other Factor, or $(G\times H)/H\approx G$
- $\langle a^i \rangle = \langle a^j \rangle$ iff $\gcd(i,n)=\gcd(j,n)$
- $\langle a^k\rangle = \langle a^{\gcd(k,|a|)}\rangle$
- $\mathbb{Z}_n$ and $U(n)$ are groups
- $\text{Inn}(G)$ is a normal subgroup of $\text{Aut}(G)$
- $G/Z(G)$ is isomorphic to $\text{Inn}(G)$
- $R/A$ Is A Field Iff $A$ Is Maximal
- $R/A$ Is An Integral Domain Iff $A$ is prime
- $|a^k| = |a|/\gcd(k,|a|)$
- A homomorphism is an $n$-to-one map
- Abelian Groups Have An Element Of Order $p$ For Every Prime That Divides The Order Of The Group
- A Direct Product Of Cyclic Groups Is Cyclic If And Only If The Groups Have Relatively Prime Orders
- Adjoining The Square Root Of A Positive Integer To $\mathbb{Z}_p$
- A Factor Ring Of The Gaussian Integers
- Affine Encryption Key Introduction
- A Homomorphism From Z To A Ring With Unity
- Algebraic Numbers
- Algebraic Properties Of 3 D Vectors And The Cross Product
- Algebraic Properties Of Modular Arithmetic
- Algebraic Properties Of Polynomial Rings Over A Field
- Algebraic Properties Of Square Matrices
- A Matrix Factor Ring
- An Isomorphism From $U(st)$ To $U(s)\oplus U(t)$ When $s$ And $t$ Are Relatively Prime
- An Ascending Chain Of Ideals In The Integers
- An External Direct Product Of Subgroups Is A Subgroup Of An External Direct Product
- An Infinite Cayley Graph
- A Polynomial Ring Over An Integral Domain Is An Integral Domain
- Ascending Chains of Ideals In a PID Are Finite
- Automorphisms Of A Directed Square
- Automorphisms Of A Square
- Automorphisms Of A Square 2
- Automorphisms On Several Graphs With 4 Vertices
- Binary Operation Introduction
- Cancellation Laws For Groups
- Can We Use Division To Create A Group
- Cayley Graphs and Isomorphisms between $\mathbb{Z}_n$ and $U(m)$
- Cayley Graph Patterns
- Cayley Graphs From Relations
- Cayley Graphs Of External Direct Products Of Cyclic Groups
- Cayley Graphs Of Two Automorphism Groups
- Cayley Tables And Isomorophisms On A Group Of Order 4
- Characteristics And Rings With Unity
- Characterizing Closed Sets Of Permutations
- Characterizing When Factor Rings Are Fields
- Closed Under Function Composition
- Composing Permutations Using Disjoint Cycle Notation
- Computing Multiplicative Inverses For Small N
- Computing Powers Modn Conjecture
- Conjecturing The Order Of General Linear Groups
- Conjugacy Is An Equivalence Relation
- Connecting Multiples And Spans Of Integers
- Continue Working On Open Problems
- Coset Products Of The Automorphisms Of The Square
- Creating Cayley Graphs Of Simple Shift Permutations
- Cryptography Reading Assignment
- Disjoint Cycle Notation Practice With Automorphisms Of A Square
- Division Algorithm For Polynomials Proof
- Division Algorithm Proof
- Do We Need The Associative Law
- Eisenstein's Criterion
- Every Element is a Product of Irreducibles in a PID
- Every Field Contains A Subfield Isomorphic To $\mathbb{Z}_p$ or $\mathbb{Q}$
- Every Finite Cyclic Group Of Order $n$ Is Isomorphic To $\mathbb{Z}_n$
- Every Infinite Cyclic Group Is Isomorphic to $\mathbb{Z}$
- Every Polynomial Has A Root In Some Extension Field
- Every Disjoint Cycle Can Be Written As A Product Of Transpositions
- Examples Of Rings Different Than The Integers
- Existence Of A Splitting Field
- External Direct Products Of Abelian And Cyclic Groups
- Factor Rings Of $\mathbb{Z}$ And $\mathbb{Z}[x]$
- Factor Groups Preserve Being Cyclic And Abelian
- Factoring By An Irreducible Gives A Simple Extension
- Fermat's Little Lemma
- Finding Idempotent And Nilpotent Elements In A Matrix Ring
- Finite Integral Domains Are Fields
- Finite Subgroup Test
- First Encryption Key
- First Isomorphism Theorem For Rings Proof
- Function Composition Is Associative
- Gaussian Integers Under Modular Arithmetic
- General Linear Group Introduction
- Groups Of Order $p^2$ Are Abelian
- Groups Of Order $pq$
- Heisenberg Matrix Group
- Homomorphisms Preserve Normal Subgroups
- Homomorphisms Preserve The Identity And Inverses
- How Do You Build A Multiplication Table From A Cayley Graph
- Ideals Give Us Factor Rings
- Identification Graphs Using Normal Subgroups Are Cayley Graphs
- If A Factor Group $G/N$ Has An Element Of Order $k$ Then So Does $G$
- If The Factor Group Of G By The Center Is Cyclic Then G Is Abelian
- Images Of Abelian And Cyclic Groups
- Induction
- Integer Linear Combination Practice
- Integral Domains Have The Cancellation Law
- Internal Direct Products Are Isomorphic To External Direct Products
- Introduction To Cayley Graph Isomorphisms
- Introduction To Internal Direct Products
- Inverses In Groups
- Inverting Function Composition
- Irreducible Polynomials Have A Zero In Some Extension Field
- Irreducibles Behave Like Prime Numbers
- Isomorphisms Yield An Equivalence Relation On The Set Of All Groups
- Kernels Are Closed Under Multiplication By Arbitrary Elements
- Lagrange's Theorem Proof
- Listing The Properties Of A Field
- Matrix Encryption Mod 5
- Mod P Irreducibility Test
- Modular Arithmetic Properties
- Multiple Zeros and The Derivative Proof
- Not Every Subring Is An Ideal
- Number Of Permutations
- Obtaining A New Ideal By Adding One Element
- One Step Subgroup Test
- Orders Of $\mathbb{Z}_n$ And $U(n)$ And Their Elements
- Permutations of $U(n)$
- Permutation Scoring On S 3
- Polynomial Rings Over PIDs need not be PIDs
- Polynomials of degree $n$ have at most $n$ zeros
- Powers Of Products In An Abelian Group
- Powers With Modular Arithmetic
- Practice With Homomorphisms from $\mathbb{Z}_n$ to $\mathbb{Z}_d$
- Practice With Cyclic Subgroups
- Practice With Identification Graphs On An Abelian Group
- Practice With Identification Graphs On A Non Abelian Group
- Practice With Set Products
- Prime Implies Irreducible
- Problem.Practice With Identification Graphs Of $\mathbb{Z}$
- Properties Of $\langle a \rangle$ When $a$ Has Finite Order
- Properties Needed For A Factor Group To Be A Factor Ring
- Properties Of Closed Sets Of Permutations
- Properties Of Cosets
- Properties Of Cosets Part Two
- Properties Of Ring Homomorphisms
- Rational Root Test
- Recognizing Rings
- Reducibility Over Q Implies Reducibility Over Z
- Reducibility Test For Degrees 2 And 3
- Relationship Between S And Its Span
- Remainders Equal Iff Difference Is A Multiple
- Ring Elements And Integer Multiplication
- Rules Of Multiplication
- Showing The Existence of a Field Extension Generated by a subset
- Simple Extensions For The Same Polynomial Are Isomorphic
- Simple Matrix Encryption
- Simple Shift Repetition
- Simple Shift Repetition Game
- Some Polynomial Factor Rings Of $\mathbb{Z}[x]$
- Some Normal Subgroups
- Some Polynomial Factor Rings
- Some Subrings Of The Complex Numbers
- Spans Of Permutations Are Subgroups
- Subgroups And Normality
- Subgroups Cosets And Identification Graphs Of The Automorphisms Of The Square
- Subgroups Of A Quotient Group Correspond To Subgroups Of The Original Group
- Subgroups Of Cyclic Groups Are Cyclic
- Subgroups Of Index 2 Are Normal
- Subring Test
- The Intersection Of Two Subgroups Of $\mathbb{Z}$
- The order of $a$ is a multiple of the order of $f(a)$, or $|a|=k|f(a)|$
- The Quotient Group $G/N$ Is A Group
- The set $U_d(n)$ and the homomorphism from $U(n)$ to $U(d)$
- The Set Product $Ha$ Preserves Determinants
- The Subgroup Test - Subgroups Are Subsets That Are Closed Under Products And Taking Inverses
- The Sum and Product Rule for Derivatives
- The Alternating Group Is A Subgroup Of The Symmetric Group
- The Center Of A Dihedral Group
- The Center Of Group Is A Subgroup
- The Class Equation
- The Collection Of Nonempty Subsets Of A Group Is A Monoid
- The Composition Of Permutations Is A Permutation
- The Degree Of A Product Of Polynomials
- The Determinant Map Is A Homomorphism
- The Factor Theorem
- The Field Of Quotients Of An Integral Domain
- The First Isomorphism Theorem Proof
- The First Sylow Theorem Proof
- The Game Of Permutation Scoring On A Square
- The Game Of Scoring
- The Game Of Scoring Misere
- The Gaussian Integers Is An Integral Domain
- The GCD Theorem Proof
- The Ideal Generated By By A Subset Is An Ideal
- The Ideal Test
- The Identity And Inverses Are Unique
- The Image Of A Subgroup Under A Homomorphism Is A Subgroup
- The Intersection Of Any Nonempty Collection Of Subgroups Is A Subgroup
- The Intersection Of Two Subgroups
- The Inverse In A Finite Group Is A Power Of The Element
- The Kernel Of A Homomorphism Is A Subgroup
- The Normal Subgroup Test
- The Number Of Conjugates Of A
- The Number Of Elements In A Set Product
- The Order Of A Finite Field
- The Order Of An Element Divides The Order Of A Group
- The Order Of An Element In An External Direct Product Is The Least Common Multiple Of The Orders Of The Elements
- The Orders Of Elements Match In Isomorphic Groups
- The Point Group Of A Cube
- The Preimage Of A Subgroup Under A Homomorphism Is A Subgroup
- The Rational Are Obtained From The Integers From An Equivalence Relation
- The Remainder Theorem
- The Right And Left Cosets Of The Kernel Are Equal
- The Set Of Simple Shift Permutations
- The Set Product Is A Binary Operation On Cosets Of Normal Subgroups
- The Span Of A Set Of Integers Is Closed
- The Span Of A Set Of Permutations Is Closed
- The Span Of A Simple Shift
- The Special Linear Group Is A Subgroup Of The General Linear Group
- The Subgroup Generated By An Element Is Actually A Subgroup
- The Subgroup Generated By S Equals The Span Of S
- The Subgroup Generated By S Is Actually A Subgroup
- The Union Of Two Subgroups
- Three Similar Cayley Graphs From Different Contexts
- Translating By An Arrow In A Cayley Graph Is An Automorphism
- Unique Factorization In A PID
- Unique Factorization Existence Proof
- Unique Factorization Uniqueness Proof
- Visualizing Cosets In A Cayley Graph
- We have $I=\left<g(x)\right>$ if and only if $g(x)$ is a polynomial of minimal degree in $I$
- We Know $\left<p(x) \right>$ Is Maximal Iff $p(x)$ Is Irreducible
- What Properties Do Subrings Inherit
- When Does $H=Ha$
- When Is $HK$ A Subgroup Of $G$
- When is $x \pmod d$ a homomorphism from $\mathbb{Z}_n$ to $\mathbb{Z}_d$
- When Does An Integer Have A Modular Multiplicative Inverse
- When Do Two Simple Shifts Span The Same Set
- When Is A Polynomial Factor Ring An Integral Domain
- Which Groups Of Order 60 Are Isomorphic
- Why Are Zero Divisors Such A Problem
Alphabetical List of Exercises
- A Homomorphism From $\mathbb{Z}_n$ to $\mathbb{Z}_d$ when $d$ is a divisor of $n$
- A Direct Product Of Cyclic Groups Is Cyclic If And Only If The Groups Have Relatively Prime Orders
- Are The Natural Numbers A Subgroup Of The Integers
- A Subgroup Is Normal If And Only If It Is The Kernel Of Some Homomorphism
- Centralizers Of The Automorphisms Of The Square
- Conjugacy Classes Of The Automorphisms Of The Square
- Cyclic Groups Are Abelian
- Even Permutations
- If $a^k=e$, Then The Order Of $a$ Divides $k$
- Order Is The Smallest Positive Integer
- Practice With Cosets Of $3\mathbb{Z}$
- Practice With Even Permutations
- Practice With Order
- Practice With Orders In External Direct Products
- Properties Of An Element With Infinite Order
- Simple Matrix Encryption
- The External Direct Product $G\oplus H$ is a Group of order $|G||H|$
- The Order Of $U(p)$ When $p$ Is Prime
- The set of simple shift permutations on 26 letters is isomorphic to $\mathbb{Z}_{26}$.
- The Subgroups Of $\mathbb{Z}$ are $n\mathbb{Z}$
- The Symmetric Group of Degree $n$ Is A Group
- The Integers Under Addition Are Isomorphic To The Positive Integers Under Multiplication
- The Last Element Of UN Is Always Its Own Inverse
- The Order Of A In UP Divides P-1
- The Order Of An Inverse
- What Is The Group Operation On The Integers
Most Recently Modified Problems
- A homomorphism is an $n$-to-one map
- The Order Of An Element Divides The Order Of A Group
- The order of $a$ is a multiple of the order of $f(a)$, or $|a|=k|f(a)|$
- Some Normal Subgroups
- Lagrange's Theorem Proof
- The Set Product Is A Binary Operation On Cosets Of Normal Subgroups
- Practice With Identification Graphs On A Non Abelian Group
- Homomorphisms Preserve The Identity And Inverses
- The Set Product $Ha$ Preserves Determinants
- Practice With Identification Graphs On An Abelian Group