# Study sets matching "abstract algebra"

Study sets

Diagrams

Classes

Users

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

Mathematical Proof

Hypothesis

Conclusion

Proposition

is a convincing argument about the accuracy of a statement

a proposition made as a basis for reasoning, without any assum…

a judgment or decision reached by reasoning

if we can prove a statement true, then the statement is called…

Mathematical Proof

is a convincing argument about the accuracy of a statement

Hypothesis

a proposition made as a basis for reasoning, without any assum…

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…

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…

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…

Cayley's Theorem

Normal Subgroup Test

Cauchy's Theorem for Finite Abelian Gro…

|A|

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

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

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 on…

Normal Subgroup Test

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

Group

Abelian Group

Subgroup ( Notation H<=G)

Order of a Group

A group G is a set of objects with a binary operation satisfyi…

A group which is commutative. That is, for any elements x,y in…

A subset H of G is called a subgroup if: ... a) for every x,y in…

The number of elements in the Group

Group

A group G is a set of objects with a binary operation satisfyi…

Abelian Group

A group which is commutative. That is, for any elements x,y in…

well ordering principle

divisor

prime

greatest common divisor

every nonempty set of positive integers contains a smallest me…

a non-zero integer t is a _______ of an integer s if there is…

a positive integer greater than 1 whose only positive divisors…

the ______ of two non-zero integers a and b is the largest of…

well ordering principle

every nonempty set of positive integers contains a smallest me…

divisor

a non-zero integer t is a _______ of an integer s if there is…

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|.…

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…

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…

Divisibility

Greatest Common Divisor

Prime Number

Composite number

Let a and b be integers, b/=0. It is said that a is divisible…

Let a,b bel Z, |a|+|b|>0.... A number is said to be GCD of a and…

An integer p is said to be prime provided that p/=1,-1,0 and i…

Any integer n, |n|>1, that is not prime

Divisibility

Let a and b be integers, b/=0. It is said that a is divisible…

Greatest Common Divisor

Let a,b bel Z, |a|+|b|>0.... A number is said to be GCD of a and…

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…