# Study sets matching "abstract algebra"

Group

Cyclic

Order

Subgroup

A pair (G, *) such that... 1) G is a set and * is a binary operat…

A group is called this if there is an element x in G such that…

The number of elements in a group G, denoted |G|

A subset of a group G that satisfies:... 1) h1*h2 is in H for all…

Group

A pair (G, *) such that... 1) G is a set and * is a binary operat…

Cyclic

A group is called this if there is an element x in G such that…

Order of a group

Order of an element

Index of a subgroup H in a group G

Partition

Number of elements in a group

Smallest number of times so that g^n=ggg...=e

The number of cosets of H in G (G:H)

If G acts on A, orbits partition A, or their union is A, and c…

Order of a group

Number of elements in a group

Order of an element

Smallest number of times so that g^n=ggg...=e

Division algorithm

GCD is a linear combination

fundamental theorem of Arithmetic

first principle of mathematical inducti…

let a and b be integers with b greater than 0 then there exist…

For any nonzero integers a and b, there exist integers s and t…

every positive integer n>1 is either a prime or a product of p…

If (i) P(a) is true for the starting point a which is a positi…

Division algorithm

let a and b be integers with b greater than 0 then there exist…

GCD is a linear combination

For any nonzero integers a and b, there exist integers s and t…

Proper Subset

Cartesian Product

Domain and Range of A-> B

Surjective or Onto

B is a subset of A but they are NOT equal

AxB {(a,b) :a elem of A/ b elem of B}

Domain A... Range B

if f: A->B is a map and the image of f is B then f is surjecti…

Proper Subset

B is a subset of A but they are NOT equal

Cartesian Product

AxB {(a,b) :a elem of A/ b elem of B}

Well Ordering Principle

Division Algorithm

GCD is a Linear Combination

GCD corollary

Every nonempty set of positive integers contains a smallest me…

Let a and b be integers with b>0. Then there exists unique int…

For any nonzero integers a and b, there exists integers s and…

If a and b are relatively prime, then there exists integers s…

Well Ordering Principle

Every nonempty set of positive integers contains a smallest me…

Division Algorithm

Let a and b be integers with b>0. Then there exists unique int…

binary operation

binary structure

Associative

semigroup

a nonempty set S, together with a binary operation defined on…

...

binary operation

binary structure

a nonempty set S, together with a binary operation defined on…

parts of a =bq ; b|a

aZ =

Well-Ordering Principle

Divison Algorithm

a is a multiple, b is a divsor (factor), q is some integer

= { m ∈ Z | m=aq for some q ∈ Z }

Every nonempty set of natural numbers contains a smallest elem…

For any a, b, there exists q, r such that a=bq+r

parts of a =bq ; b|a

a is a multiple, b is a divsor (factor), q is some integer

aZ =

= { m ∈ Z | m=aq for some q ∈ Z }

set

well defined set

subset of a set

improper subset/proper subset

1. made up of elements... 2. the empty set is the only set with n…

if S is a set and a is some object, then either a is definitel…

A is a subset of B if every element of B is in A ... (B does not…

A is a improper subset of A... any subset besides A is a proper s…

set

1. made up of elements... 2. the empty set is the only set with n…

well defined set

if S is a set and a is some object, then either a is definitel…

normal subgroup

quotient group

homomorphism

kernels

Let G be a group, H a subgroup. H is a normal subgroup if for…

Let G be a group, H a normal subgroup. G/H is the set of left…

Let G1, G2 be two groups. Then a homomorphism from G1 to G2 is…

The kernel of a homomorphism phi:G1->G2 is the preimage of e(g…

normal subgroup

Let G be a group, H a subgroup. H is a normal subgroup if for…

quotient group

Let G be a group, H a normal subgroup. G/H is the set of left…

Dihedral Group... *order=

Cyclic Group... *order

Compositions of rotations/refelections

Definition of a group

Dn... Rotations and Reflections... order=2n

Zn or Sn... Rotations... order=n

rotation=1... reflection=-1

1. Closure--for all a,b in G, ab in G... 2. Identity--there is an…

Dihedral Group... *order=

Dn... Rotations and Reflections... order=2n

Cyclic Group... *order

Zn or Sn... Rotations... order=n

Ring

S is a subring of R iff

A Ring is commutative iff

Integral domain

(R,+) is an abelian group... * is associative and has an identity…

1. 0r and 1R are in S... 2. Closure over +,*... 3. If x is in S then…

(R,*) is commutative

R is commutative... 0 =/= 1... xy = 0 => x = 0 or y = 0

Ring

(R,+) is an abelian group... * is associative and has an identity…

S is a subring of R iff

1. 0r and 1R are in S... 2. Closure over +,*... 3. If x is in S then…

integral domain

The Division Algorithm for F[x], where…

Remainder Theorem

Factor Theorem

Let (R, +, ·) be a commutative ring with 1. The ring (R, +, ·)…

Let F be a field, and p(x) and a(x) be polynomials in F[x]. If…

Let F be a filed. Let s 2 F and p(x) 2 F[x]. Then there is a u…

Let F be a filed. Let p(x) 2 F[x] and s 2 F. s is a zero of p(…

integral domain

Let (R, +, ·) be a commutative ring with 1. The ring (R, +, ·)…

The Division Algorithm for F[x], where…

Let F be a field, and p(x) and a(x) be polynomials in F[x]. If…

GCD as a linear combination

One-Step Subgroup Test

Two-Step subgroup test

Finite subgroup Test

Let a and b be positive integers. Then there exist integers s…

Let G be a group and H a non-empty subset of G. If ab-1 whenev…

If H is a non-empty subset of G and whenever a,b, E H we know…

If H is a finte, non=empty subset of a group G, and ab E G whe…

GCD as a linear combination

Let a and b be positive integers. Then there exist integers s…

One-Step Subgroup Test

Let G be a group and H a non-empty subset of G. If ab-1 whenev…

group action

homomorphism

<g> (g in angle brackets)

s bar

Let G be a group, A be a set. A group action of G on A is map…

Let (G, star) and (H, square) be 2 groups. Then a map, phi, fr…

a cyclic subgroup generated by g

Let G be a general group, and S a subset of G. The subgroup ge…

group action

Let G be a group, A be a set. A group action of G on A is map…

homomorphism

Let (G, star) and (H, square) be 2 groups. Then a map, phi, fr…

closed under the operation

closed under taking the inverses

Subgroup

Order of a Group

if ab is in H and whenever a and b are in H

a inverse is in H whenever a is in H

if the subset H of group G is itself a group under the operati…

the number of elements of a group (infinite or finite) is call…

closed under the operation

if ab is in H and whenever a and b are in H

closed under taking the inverses

a inverse is in H whenever a is in H

Well-ordering Principle

Greatest common divisor of a and b

THM: gcd(a,b)= MIN{

A partition of a set S is

Every non empty set of positive integers has a least element.

The greatest integer d such that d divides a and d divides b

ax+by such that x,y in Z and ax+by>0

collection of nonempty disjoint subsets of S whose union is S

Well-ordering Principle

Every non empty set of positive integers has a least element.

Greatest common divisor of a and b

The greatest integer d such that d divides a and d divides b

Fundamental Theorem of Arithmetic

Euclid's Lemma

GCD Corollary

GCD is a Linear Combination

Every integer greater than 1 is a prime or a product of primes…

If p is a prime that divides ab then p/a or p/b.

If a and b are relatively prime, then there exist integers s a…

For any nonzero integers a and b there exist integers s and t…

Fundamental Theorem of Arithmetic

Every integer greater than 1 is a prime or a product of primes…

Euclid's Lemma

If p is a prime that divides ab then p/a or p/b.

Well Ordering Principle

Division Algorithm

Greatest Common Divisor / Relative Prim…

GCD is a Linear Cmbination

Every nonempty set of positive integers contains a smallest me…

Let a and b be integers with b > 0. Then there exist unique in…

The greatest common divisor of two nonzero integers a and b is…

For any nonzero integers a and b, there exists integers s and…

Well Ordering Principle

Every nonempty set of positive integers contains a smallest me…

Division Algorithm

Let a and b be integers with b > 0. Then there exist unique in…

Groups

Binary operation

Commutative

Associative

An abstraction of the integers under addition

Let S be a set. A binary operation

**is a function**: SxS -> SLet

**be a binary operation on S. Then**is commutative if for…Let

**be a binary operation on S. Then**is associative if for…Groups

An abstraction of the integers under addition

Binary operation

Let S be a set. A binary operation

**is a function**: SxS -> SDEFINITION: Set equivalence (A = B)

DEFINITION: Subset (A ⊆ B)

DEFINITION: Empty set (Ø)

DEFINITION: Power set

Two sets, A and B, are equal (A=B) if and only if they contain…

For all a∈A, we also have a∈B.

The set that does not contain any elements. It is a subset of…

The set of subsets.

DEFINITION: Set equivalence (A = B)

Two sets, A and B, are equal (A=B) if and only if they contain…

DEFINITION: Subset (A ⊆ B)

For all a∈A, we also have a∈B.

binary structure

semigroup

monoid

group

a nonempty set with a binary operation defined on it

a binary structure whose binary operation is associative

A semigroup that has an identity is called a monoid

A monoid, where every element has an inverse.

binary structure

a nonempty set with a binary operation defined on it

semigroup

a binary structure whose binary operation is associative

Commutative

Identity element

Inverse

Operation

Independent of order ab=ba

a

**e=a and e**a=a for every a in Aa

**a^(-1)=e and a^(-1)**a=e for every a in ALet A be any set. An operation on A has the following aspects…

Commutative

Independent of order ab=ba

Identity element

a

**e=a and e**a=a for every a in A