# Study sets matching "abstract algebra"

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Lagrange's theorem

Left regular representation of a group G

Cosets

Proving isomorphism

if G is a finite group and H is a subgroup of G then the order…

is a map phi from G into Sg for g that is an element of G the…

Let G be a group and let H be a nonempty subset of G. For any…

define a function phi from G to G(bar), prove phi is one to on…

Lagrange's theorem

if G is a finite group and H is a subgroup of G then the order…

Left regular representation of a group G

is a map phi from G into Sg for g that is an element of G the…

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The Well Ordering Property

The Division Algorithm

Relatively Prime Integers

Euclid's Lemma

For every non empty set of positive integers, there exists a s…

For all integers a and b, there exists unique integers q and r…

a and b are called relatively prime provided gcd(a,b) = 1.

For integers a and b and prime integer p, if p divides (ab) th…

The Well Ordering Property

For every non empty set of positive integers, there exists a s…

The Division Algorithm

For all integers a and b, there exists unique integers q and r…

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

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…

What is the first aspect of an operator?

What is the second aspect of an operato…

What is the third aspect of an operator?

What is the formal definition of an ope…

a*b is defined for every ordered pair (a,b) of elements of a s…

a*b must be uniquely defined, I.e. they should be unambiguous.

If a and b are in A, a*b must be in the set A.

An operation * on a given set A is a rule which assigns to eac…

What is the first aspect of an operator?

a*b is defined for every ordered pair (a,b) of elements of a s…

What is the second aspect of an operato…

a*b must be uniquely defined, I.e. they should be unambiguous.

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 }

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…

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…

The Well-Ordering Principle

a divides b/a is a divisor of b if

The greatest common divisor denoted by…

a,b are relatively prime if

Every nonempty set of positive integers has a least element

b=ak

the largest integer d that divides both a and b

gcd(a,b) is 1

The Well-Ordering Principle

Every nonempty set of positive integers has a least element

a divides b/a is a divisor of b if

b=ak

Greatest Common Divisor

gcd (q(x), r(x))

irreducible

C[x]. (x + i)(x - i)

Let f(x) and g(x) be polynomials in R[x] with at least one not…

d(x) is a gcd (f(x), g(x)) if and only if d(x) is a

Let f(x) e R[x] where R is an integral domain. f(x) is said to…

f(x) = x^2 +1 is reducible in ________.

Greatest Common Divisor

Let f(x) and g(x) be polynomials in R[x] with at least one not…

gcd (q(x), r(x))

d(x) is a gcd (f(x), g(x)) if and only if d(x) is a

Well ordered set

Proof by induction

Fibonacci numbers

Definition of divides

A set where every non empty subset has a smallest element- gua…

Let P(n) be a statement about the natural numbers n.... If the fo…

Fn + Fn+1 = Fn+2

Let a and b be 2 nonzero integers. we say b divides a and writ…

Well ordered set

A set where every non empty subset has a smallest element- gua…

Proof by induction

Let P(n) be a statement about the natural numbers n.... If the fo…

Binary Operation

Group

Order of a Group

Order of an Element

A function that assigns each ordered pair in G another element…

A set G with binary operation. If associativity, inverses and…

The number of elements in a group.

The smallest possible integer, n, such that g^n=e. If not n ex…

Binary Operation

A function that assigns each ordered pair in G another element…

Group

A set G with binary operation. If associativity, inverses and…

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Cosets

Poperties of cosets

Lagrange's Thm

Index

G is a group, H c G... For all a in G, define aH = {ah | h E H} a…

a E aH... AH = H iff a E H... AH = bH or aH (Union) bH = the empty s…

If G is a finite group and H < G, then |H| divides |G|.... Order…

The index of H in G, written |G:H| is the number of left or ri…

Cosets

G is a group, H c G... For all a in G, define aH = {ah | h E H} a…

Poperties of cosets

a E aH... AH = H iff a E H... AH = bH or aH (Union) bH = the empty s…

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…

Theorem 4.2... Relating |a| and <a^k>

Strategy for showing two things are equ…

Corollary 1 of Theorem 4.2... Cyclic group…

Corollary 2 of Theorem 4.2... <a_i> = <a_j>

Let a be an element of order n in a group G. Let k(in)Z^+. The…

Show they are subgroups of each other

In a finite cyclic group, the order of an element divides the…

Let a be an element of G and |a| = n.... Then, <a^i> = <a^j> iff…

Theorem 4.2... Relating |a| and <a^k>

Let a be an element of order n in a group G. Let k(in)Z^+. The…

Strategy for showing two things are equ…

Show they are subgroups of each other

Ring

Unity

Unit

Thm 12.1 - 6 properties of multiplicati…

A ring R is a set with two binary operations, addition (a+b) a…

A nonzero element that is an identity under multiplication.

A nonzero element of a commutative ring with a multiplicative…

1. a0 = 0a = 0 ... 2. a(-b) = (-a)b = -(ab)... 3. (-a)(-b) = ab... 4. a…

Ring

A ring R is a set with two binary operations, addition (a+b) a…

Unity

A nonzero element that is an identity under multiplication.