Abstract Algebra Final

Well Ordering Principle

Every nonempty set of positive integers contains a smallest member

Equivalence Relation

A set of ordered pairs of elements of S st 1) (a,a) \\in R for all a \\in S (reflexive) 2) (a,b) \\in R implies (b,a) \\in R (symmetric) 3) (a,b) \\in R and (b,c) \\in R implies (a,c) \\in R (transitive)

equivalence class

If \~ is an equivalence relation on a set S and a \\in S, then the set \[a\]={x\\in S | x squiggly a}

partition

A partition of a set S is a collection of nonempty disjoint subsets of S whose union is S

binary operation

a function that assigns each ordered pair of elements of G an element of G.

group

Let G be a set together with a binary operation (usually called multiplication) that assigns to each ordered pair (a, b) of elements of G an element in G denoted by ab. We say G is this under this operation if the following three properties are satisfied.


1. Associativity. The operation is associative; that is, (ab)c= a(bc) for all a, b, c in G.
2. Identity. There is an element e (called the identity) in G such that ae=ea=a for all a in G.
3. Inverses. For each element a in G, there is an element b in G (called an inverse of a) such that ab=ba=e.

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.

order of an element

the smallest positive integer n such that g^n=e. (In additive notation, this would be ng= 0.) If no such integer exists, we say that g has infinite order. The order of an element g is denoted by |g|.

subgroup

If a subset H of a group G is itself a group under the operation of G

cyclic subgroup

Let G be a group, and let a be any element of G. Then, is a subgroup of G. ={a^n|n \\in Z}

center of a group

the subset of elements in G that commute with every element of G. In symbols, Z(G)={a \\in G | ax= xa for all x in G}

permutation

a function from A to A that is both one to-one and onto.

permutation group

a set A is a set of permutations of A that forms a group under function composition.

dihedral group

a group of symmetries of a regular n-gon

group isomorphism

An isomorphism \\phi from a group G to a group G is a one-to-one mapping (or function) from G onto G that preserves the group operation. That is, \\phi(ab) = \\phi(a)\\phi(b) for all a, b in G. If there is an isomorphism from G onto G, we say that G and G are isomorphic and write G \\cong G.

automorphism

An isomorphism from a group G onto itself

inner automorphism

Let G be a group, and let a \\in G. The function \\phi_a defined by \\phi_a(x) = axa^-1 for all x in G

coset

Let G be a group and let H be a nonempty subset of G. For any a \\in G, the set {ah | h \\in H} is denoted by aH

normal subgroup

A subgroup H of a group G is called a normal subgroup of G if aH =Ha for all a in G.

quotient group

When the subgroup H of G is normal, then the set of left (or right) cosets of H in G is itself a group.

\
Let G be a group and let H be a normal subgroup of G. The set G/H = {aH | a \\in G} is a group under the operation (aH)(bH)=abH

group homorophism

a group G to a group G is a mapping from G into G that preserves the group operation; that is, \\phi(ab) =\\phi(a)\\phi(b) for all a, b in G

kernel of a homorphism

a group G to a group with identity e is the set {x \\in G | \\phi(x) =e}

ring

a set with two binary operations, addition (denoted by a + b) and multiplication (denoted by ab), such that for all a, b, c in R:


1. a + b = b + a.
2. (a + b) + c = a + (b + c).
3. There is an additive identity 0. That is, there is an element 0 in R such that a + 0 = a for all a in R.
4. There is an element -a in R such that a + (-a) = 0.
5. a(bc) = (ab)c.
6. a(b + c) = ab + ac and (b + c) a = ba + ca

subring

A subset S of a ring R is a if S is itself a ring with the operations of R.

zero-divisors

a nonzero element a of a commutative ring R such that there is a nonzero element b \\in R with ab= 0

integral domain

a commutative ring with unity and no zero-divisors

field

a commutative ring with unity in which every nonzero element is a unit.

characteristic of a ring

the least positive integer n such that nx = 0 for all x in R. If no such integer exists, we say that R has characteristic 0

nilpotent

Let a belong to a ring R with unity and suppose that a^n = 0 for some positive integer n

idempotent

A ring element a is called an idempotent if a^2= a