#### Study sets matching "abstract algebra"

#### Study sets matching "abstract algebra"

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…

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

equivalence relation

Greatest common divisor on z

isomorphism between groups

group

a relation that satisfies three properties for all x,y,z in s.…

let r and s be two positive integers. the positive generator o…

a function between two groups that sets up a one-to-one corres…

a set, closed under binary operation that is associative, iden…

equivalence relation

a relation that satisfies three properties for all x,y,z in s.…

Greatest common divisor on z

let r and s be two positive integers. the positive generator o…

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 Operations

Operation Properties

Group

Abelian

Defined over all ordered pairs;... Unique result;... If operands are…

Commutative;... Associative;... Identity element;... Inverse

A set G and operation * that is associative, has a neutral ele…

Group G with operation * is commutative.

Binary Operations

Defined over all ordered pairs;... Unique result;... If operands are…

Operation Properties

Commutative;... Associative;... Identity element;... Inverse

1st Sylow Theorem

2nd Sylow Theorem

3rd Sylow Theorem

Burnside's Counting Theorem

Let G be a finite set and p a prime such that p^r divides |G|.…

Let G be a finite group and p a prime such that p divides |G|.…

Let G be a finite group and p a prime where p divides |G|. The…

Let X be a G-set and G be a finite group. Then let k be the nu…

1st Sylow Theorem

Let G be a finite set and p a prime such that p^r divides |G|.…

2nd Sylow Theorem

Let G be a finite group and p a prime such that p divides |G|.…

Binary Operation * on a set S

Commutative operation

Associative soperation

A group (G, *)

is a function that associates to each order pair (s1, s2) of e…

s1

**s2 = s2**s1 for every s1, s2 E S(s1

**s2)**s3= s1**(s2**s3) for every s1,s2,s3 E S1)G is a set and * is a binary operation on G... 2)* is associati…

Binary Operation * on a set S

is a function that associates to each order pair (s1, s2) of e…

Commutative operation

s1

**s2 = s2**s1 for every s1, s2 E SCommutative

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 AWell-Ordering Axiom

Divisbility

Greatest Common Divisor (gcd)

Relatively Prime

Every nonempty subset of the set of nonnegative integers conta…

Let a and b be integers with b =/ 0. We say b|a or b is a fact…

Let a and b be integers, not both 0. The gcd of a and b is the…

Two integers whose greatest common divisor is 1

Well-Ordering Axiom

Every nonempty subset of the set of nonnegative integers conta…

Divisbility

Let a and b be integers with b =/ 0. We say b|a or b is a fact…

Associativity

The Well-Ordering Priciple

injectivity

surjectivity

addition and multiplication of real numbers is... (x+y)+z = x + (…

Let S be a nonempty subset of the natural numbers. Then with t…

S,T sets, f:S to T f is injective (or 1 to 1) if f(s_1)=f(s_2)…

S,T sets, f:S to T f is surjective (or onto) if for all t in T…

Associativity

addition and multiplication of real numbers is... (x+y)+z = x + (…

The Well-Ordering Priciple

Let S be a nonempty subset of the natural numbers. Then with t…

Ring

Additive identity

Additive inverses

Theorem1

A set A with operations called "addition" and "multiplication"…

"Zero"

"Negatives"

Let a,b be in a ring A ... 1) a

**0=0=0**a... 2) (-a)b=(a)(-b)=-ab... 3) (…Ring

A set A with operations called "addition" and "multiplication"…

Additive identity

"Zero"

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…