Abstract Algebra Definitions

Definition: Ring

A nonempty set 2 Closed Binary Operations 1st Operation- Abelian Group Associative, Commutative, Identity, Every element-inverse Associative under multiplication, Contains a multiplicative inverse Distributive with respect to addition

Definition: Commutative Ring

A ring is commutative if it is abelian to the second operation.

Group: SL2(R)

Special Linear Group 2x2 invertible matrices det A=1 Nonabelian

Definition: Zero Divisor

2 elements that multiply to the multiplicative identity.

Definition: Integral Domain

A Commutative Ring A Ring with a Unique Multiplicative Identity No Zero Divisors

Definition: Field

A Ring that is an Integral Domain A ring that is a Division Ring Has a Unity

Definition: Permutation

A bijection from a set to itself.

Definition: Extension Field

A field E is an extension field of F if F is a subfield of E. The field F is called the base field.

Definition: Cyclic Group

A group that can be generated by a single element There exists an element a in the group G such that, <a>=G

Definition: Abelian Group

A group that is commutative.

Definition: Homomorphism

A homomorphism between groups (G,+) and (H,*) such that φ:G-> H, φ(g1+g2)=φ(g1)*φ(g2)

Definition: Isomorphism

A mapping from one group to another that preserves operation and is a bijection.

Definition: Unity

A multiplicative Identity

Definition: Transcendental Number

A number that is not algebraic

Definition: Group

A set of objects that must be: Closed under operation, Contain an identity element Each element has an inverse Associative

Definition: Subgroup

A subdivision of a group. Has an identity Has an inverse The operation is closed

Definition: Normal Subgroup

A subgroup H of a group G if gH=Hg for all g in G.

Definition: Ideal

An ideal in a ring R is a subring I of R such that if a is in I and r is in R, then both ar and ra are in I; that is rI ⊂ I and Ir ⊂ I for all r in R.

Definition: Algebraic Number

Any element α in an extension field E over F is algebraic over F if f(α)=0 for some nonzero polynomial f(x) in F[x]. An algebraic number is one that has a polynomial which has that as a root.

Group: C*

Complex Numbers excluding 0 Multiplication Identity: 1

Group: GL2(R)

General Linear Group 2x2 invertible matrices det A n=0 Nonabelian

Definition: Direct Product G x H

If (G,+) and (H,*) are groups, and g1,g2 in G and h1,h2 in H. Then the direct product of G x H. (g1,h1)(g2,h2)= (g1+g2 , h1*h2) The operation is within the groups.

Definition: Factor Group G/N

If N is a normal subgroup of a group G, then the cosets of N in G form a group G/N, under the operation (aN)(bN)=abN

Definition: Kernel of a Homomorphism

If φ is a homomorphism φ: G -> H, the kernel of φ is the set {g∈G | φ(g) = e_{H}}.

Group: Zn

Integers mod n Addition Mod n, Abelian Identity: 0

Definition: Left Coset

Let G be a group and H be a subgroup of G. For g in G, gH={gh:h in H}.

Definition: Even/Odd Permutation

Odd- A bijection from a set to itself that has an even number of elements that can be made into a set of odd transpositions. Even- A bijection from a set to itself that has an odd number of elements that can be made into a set of even transpositions.

Group: Q(8)

Quaternians Multiplication Nonabelian Identity: 1 +-1 +-i +-j +-k

Group: R*

Real Numbers excluding 0 Multiplication Identity:1

Group: U(n)

The Natural numbers relatively prime to Z_{n} Multiplication mod n Abelian Identity: 0

Definition: Index of H in G, [G:H]

The amount of distinct left or right cosets of a subgroup H on a group G [G:H]= |G|/|H|

Definition: Order of an Element

The least positive integer n such that a^n = e. (In additive notation, this would be na = 0.) The fewest # of times when operated upon to reach identity.

Definition: Order of a Group

The number of elements of a group (finite or infinite) is called its order. We will use |G| to denote the order of G.

Group: Dn

The rigid motions of a regular n-gon. N-vertices Nonabelian Not Cyclic. Operation is composition.

Group: Z

The set of Integers Addition Abelian Cyclic Identity:0

Definition: Factor Ring

The set of all equivalence classes is denoted by R/I, thus: (a+I)+(b+I)=(a+b)+I (a+I)(b+I)=(ab)+I

Group: Sn

The set of permutations on n-objects. Composition Nonabelian Not cyclic Identity: p0