#### Study sets matching "abstract algebra"

#### Study sets matching "abstract algebra"

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 ele…

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…

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

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…

set

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

well defined set

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

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 i…

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

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…

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, f…

a cyclic subgroup generated by g

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

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, f…

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 a…

Dihedral Group... *order=

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

Cyclic Group... *order

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

DEFINITION: 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 contai…

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 contai…

DEFINITION: Subset (A ⊆ B)

For all a∈A, we also have a∈B.

Well Ordering Principle

Division Algorithm

Greatest Common Divisor / Relative Pri…

GCD is a Linear Cmbination

Every nonempty set of positive integers contains a smallest m…

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

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

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

Well Ordering Principle

Every nonempty set of positive integers contains a smallest m…

Division Algorithm

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

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 fo…Let

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

An abstraction of the integers under addition

Binary operation

Let S be a set. A binary operation

**is a function**: SxS -> SWell-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 prime…

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…

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 prime…

Euclid's Lemma

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

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…

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

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

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…

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 Aclosed 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 operat…

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

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

Binary Operations

Operation Properties

Group

Abelian

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

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

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

Group G with operation * is commutative.

Binary Operations

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

Operation Properties

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

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…

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 associat…

Binary Operation * on a set S

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

Commutative operation

s1

**s2 = s2**s1 for every s1, s2 E SDivision algorithm

GCD is a linear combination

fundamental theorem of Arithmetic

first principle of mathematical induct…

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

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

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

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

Division algorithm

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

GCD is a linear combination

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

Well-Ordering Axiom

Divisbility

Greatest Common Divisor (gcd)

Relatively Prime

Every nonempty subset of the set of nonnegative integers cont…

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

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

Two integers whose greatest common divisor is 1

Well-Ordering Axiom

Every nonempty subset of the set of nonnegative integers cont…

Divisbility

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

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…

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…

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…

Cayley's Theorem

Normal Subgroup Test

Cauchy's Theorem for Finite Abelian Gr…

|A|

Every group G is isomorphic to a (sub)group of permutations o…

A subgroup H of G is normal in G if and only if xHx⁻¹⊆H for a…

Let G be a finite Abelian group whose order is divisible by a…

The cardinality of A, the number of elements in A.

Cayley's Theorem

Every group G is isomorphic to a (sub)group of permutations o…

Normal Subgroup Test

A subgroup H of G is normal in G if and only if xHx⁻¹⊆H for a…

Well Ordering Principle

Division Algorithm

GCD is a Linear Combination

GCD corollary

Every nonempty set of positive integers contains a smallest m…

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

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 m…

Division Algorithm

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