# 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…

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

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…

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…

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 Theory

Associative binary operation

Identity element

Inverse element

The mathematic capturing of the concept of symmetry

Let G be a set. A binary operation * on G (so that the image o…

An identity element, e of G, is an element of G with the prope…

Given a binary operation * on G with identity e for each eleme…

Group Theory

The mathematic capturing of the concept of symmetry

Associative binary operation

Let G be a set. A binary operation * on G (so that the image o…

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

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…

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.

Generator

Cyclic Group

Centralizer

Subgroup

The element in the group whose order has the same number of el…

Has a generator

C(a) is the set of all elements in G in that commute with a

Satisfies Closure and Inverse rules

Generator

The element in the group whose order has the same number of el…

Cyclic Group

Has a generator

Characteristic

Lemma 16.18: Let R be a ring with ident…

Ring with Unity(Identity)

Commutative Ring

The least positive integer n such that nr=0 for all r in R. If…

The characteristic of R is n

If there is an element 1 in R such that 1!=0 and 1a=a1=a for a…

A ring for which ab=ba for all a,b in R

Characteristic

The least positive integer n such that nr=0 for all r in R. If…

Lemma 16.18: Let R be a ring with ident…

The characteristic of R is n

Order of a group

Finite group

Order of an element

Relationship between order of a group a…

The number of elements is contains

Has finite many elements

The order of an element g of a group G is the least positive i…

Order of an element always divides the order of a group

Order of a group

The number of elements is contains

Finite group

Has finite many elements

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…

subgroup

cyclic subgroup

lcm

gcd

a nonempty subset, H, of a group G is called a subgroup of G i…

a subgroup of G generated by a is a set {a^I | I any integer).…

(a\l and b\l) (if a\m and b\m then l\m)

(d\a and d\b) (if c\a and c\b then c\d)

subgroup

a nonempty subset, H, of a group G is called a subgroup of G i…

cyclic subgroup

a subgroup of G generated by a is a set {a^I | I any integer).…