Abstract Algebra (All terms)

Set

A collection of objects (elements) that are well defined, unordered, and distinct.

Empty Set

A set with no elements

Subset

Let X be a set. The set Y is a subset of X if for all y in Y, y is in X

Cardinality

The number of elements in the set

Countable

Has the same cardinality as the integers

Function

an assignment of elements in X to elements in Y such that 1. f(x) is in Y and 2. f(x) is a unique value

Domain

Set you map out of

Codomain

Set you map into

Image

The values that the function hits in the codomain

Surjective

For all y in Y there exists an x in X such that f(x)=y. Basically, you hit all things in the codomain

Injective

f(x₁ ) = f(x₂ ) if x₁ =x₂. No things go to the same thing in the codomain

Bijective

Injective and Surjective

Inverse

f ^-1 f =id=f f ^-1

Equivalence Relation

A way to group elements together that share some property such that:

1. Reflexive

2. Symmetric

3. Transitive

Equivalence Class

The set {y in X | x is equivalence relation to y}

Partition

The collection of non-empty subsets A₁ …A_n such that:

1. A = A₁ U A₂ U…U A_n (make up all of A)

2. A_i are disjoint

Singleton Partition

One element in each class

Trivial Partition

Everything in one partition

Well Ordering Principle

Every non-empty set of non negative integers has a smallest element

Division Algorithm (Theorem)

There are a uniquely determined q,r in the natural numbers st n=qd+r

Divisor

d is a divisor of n is n=qd (no remainder)

Common Divisor

If d|n and d|m

Greatest common divisor

the largest element of the set such that d|n and d|m

Coprime

gcd(m,n)= 1, then m,n are coprime

Prime

If for any d|p d=1 or d=p and p>/= 2

Euclids Lemma

Let p be prime. If p| m₁ m₂ …m_n then p|m_i for some i

Prime Factorization Theorem

Every n >/= 2 can be uniquly factored as a product of one or more primes: n= p₁ ^m₁ p₂ ^m₂ …p_k ^m_k

Square Free

for n= p₁ ^m₁ p₂ ^m₂ …p_k ^m_k all m_i =1

Congruent Modulo n

n| (a-b)

Residue Class

The equivalence class of a in the integers under mod_n

Integers Modulo n

Set of residue classes mod_n

Group

A set with a binary operation that follows:

1. G is closed under the operation

2. Operation is associative

3. Has an identity

4. Has an inverse

Abelian

A group whose operation is commutative

Cancelation Theorem

1. gh=gk then h=k

2. hg=kg then h=k

Cyclic Group

The group of rotations of a regular n-gon

Generate

All elements can be achieved via operations within the set

Subgroup

Let (G,*) be a group, A subset H of G is a subgroup if:

1. id is in H

2. H is closed under *

3. all h have an inverse in H

Permutation

A bijection on the set X_n = {1 2 … n}

Symmetric Group

The group formed by the set of permutations under the operation composition

Fixed

If f(x)=x

Disjoint Permutation

A permutation p, q in S_n is disjoint If no element in X_n is moved by both p and q

Cycle Decomposition

Let p in S_n be a non-trivial permutation. Then p can be uniquely factored as a product of cycles of length at least 2

Cycle Type

Touple that tells you:

1. The number of disjoint cycles

2. length of each cycle

Transposition

Cycle of length 2

Transposition Decomposition

Every cycle of length at least 2 can be factored as a product of transpositions

Units

Let M be a moniod. The collection of elements that have an inverse

Ring

A set with 2 binary operations st:

1. Additive Identity: 0 in R

2. Additive inverse: -r in R for all r

3. Associativity: (r+s) +u = r + (s+u) for r,s,u in R

4. Symmetric: r+s=s+r

5. Multiplicative Identity: 1 in R

6. Associative: (rs)u=r(su)

7. Distributivity: r(s+u)=rs+ru

8. Distributivity: (s+u)r= sr+ur

Commutative Rings

If the multiplication operation is commutative

Subring

Let R be a ring. S is a subset of R if:

1. Identities in S: 1 in S, 0 in S

2. S is closed under addition and multiplication operations

3. Additive Inverse: for r in S, -r is in S

Zero Product Property (ZPP)

If ab=0 then a=0 or b=0 (0 is additive id)

Integral Domain

Let R be a commutative ring. If ZPP holds, then R is an integral domain

Field

Let R be a commutative ring. R is a field if for all r not equal to 0, there is a multiplicative inverse (rr^-1=1)

Ring Homomorphism

1. f(a+b)=f(a)+f(b)

2. f(ab)=f(a)f(b)

3. f(1_R)=1_S

Ring Isomorphism

A ring homomorphism with a bijection

Ideal

Let a in R be a fixed element where R is a ring.

Ra= {ra | r in R}

aR ={ar | r in R}

An additive subgroup A is an ideal if Ra is a subgroup of A and aR is a subgroup of A

Quotient Ring

The ring R/A

Proper Ideal

An Ideal A of R st A doesnt equal R

Principle Ideal

An Ideal generated by a: <a>={ar|r in R}

Prime Ideal

Let R be a commutative Ring, an ideal P is prime if for rs in P then r in P or s in P

FIT for rings

Let f: R —> S be a ring Hom

1. Ker f is an ideal of R

2. Im f is a subring of S

3. R/Ker f is isomorphic to im f

Fundamental Theorem of Cyclic Groups

If H is a subgroup of C_n then H=C_d where d|n. Furthermore, each divisor (d) of n is the order of a unique subgroup

Group Homomorphism

Let G,H be groups and f: G →H is a map. f is a group homomorphism if f(g₁ g₂)= f(g₁) f(g₂)

Group Isomorphism

A group homomorphism f: G → H is group isomorphism if f is a bijection

Kernel

Let f: G → H be a group homomorphism: Kernel is the set {g in G | f(g)=e}

Stuff that gets mapped to the identity

Image

Let f: G → H be a group homomorphism: The image is the set {h in H | f(g)=h for some g in G}

Automorphism

A group Isomorphism from G → G is an automorphism

Inner Automorphism

Let f be a group isomorphism from G → G: f(g)= hgh^-1

Right Coset

Let H be a subgroup of G and g in G: Then Hg = {hg| h in H}

Index

Let H be a subgroup of G: The index is the number of distinct cosets of H in G

Lagrange’s Theorem

Let H be a subgroup of G with finite order: Let |H| divides |G| and the index |G:H| = |G| / |H|

Center

{h in G | gh=hg for all g in G}

Normal

1. gH=Hg for all g in G

2. ghg^-1 is in H for all g in G

Conjugate

ghg^-1

Dedekind

A group in which all subgroups are normal

Simple Group

The only normal subgroups are e and G itself

Quotient Group

Let H be normal to G: The quotient group is the group of H-Cosets

First Isomorphism Theorem

Let f: G → H be a group Homomorphism.

1. Ker f is a normal subgroup of G

2. im(f) is a subgroup of H

3. G_{/ker f} is isomorphic to im(f)