Study sets matching "abstract algebra"

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Study sets matching "abstract algebra"

45 terms
Abstract Algebra
Composition of Permutations
Order of a Permutation
Even/Odd Permutation
Permutation in Sn
Composition of Permutations
Order of a Permutation
190 terms
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 }
8 terms
Abstract Algebra
binary operation
binary structure
Associative
semigroup
a nonempty set S, together with a binary operation defined on…
...
binary operation
binary structure
a nonempty set S, together with a binary operation defined on…
Abstract Algebra
A U B
What is another name for surjective?
What is another name for injective?
What is the definition of onto or surj…
Onto
One to one
A U B
What is another name for surjective?
Onto
45 terms
Abstract Algebra
Composition of Permutations
Order of a Permutation
Even/Odd Permutation
Permutation in Sn
Composition of Permutations
Order of a Permutation
8 terms
Abstract Algebra
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…
12 terms
Abstract Algebra
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.
8 terms
Abstract Algebra
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
Abstract Algebra
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…
Abstract Algebra
Groups
Binary operation
Commutative
Associative
An abstraction of the integers under addition
Let S be a set. A binary operation is a function : SxS -> S
Let 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 -> S
Abstract Algebra
Commutative
Identity element
Inverse
Operation
Independent of order ab=ba
ae=a and ea=a for every a in A
aa^(-1)=e and a^(-1)a=e for every a in A
Let A be any set. An operation on A has the following aspects…
Commutative
Independent of order ab=ba
Identity element
ae=a and ea=a for every a in A
21 terms
abstract algebra
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…
30 terms
Abstract Algebra
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 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
Abstract Algebra
Division 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…
Abstract Algebra
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…
Abstract Algebra
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…
Abstract Algebra
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
5 terms
Abstract Algebra
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.
18 terms
Abstract Algebra
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) a0=0=0a... 2) (-a)b=(a)(-b)=-ab... 3)…
Ring
A set A with operations called "addition" and "multiplication…
Additive identity
"Zero"
157 terms
Abstract Algebra
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…
51 terms
Abstract Algebra
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
Abstract Algebra
well ordering principle
divisor
prime
greatest common divisor
every nonempty set of positive integers contains a smallest m…
a non-zero integer t is a _______ of an integer s if there is…
a positive integer greater than 1 whose only positive divisor…
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 m…
divisor
a non-zero integer t is a _______ of an integer s if there is…
30 terms
Abstract Algebra
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…
29 terms
Abstract Algebra
factor group(G/H)
automorphism
simple
maximal normal subgroup
If φ G -> G' be a group homo with Kerφ = H, then the cosets f…
an isomorphism φ:G->G of a group with itself.
nontrivial group that has no proper nontrivial normal subgrou…
is a subgroup M<G M≠G s.t. There does not exist a proper norm…
factor group(G/H)
If φ G -> G' be a group homo with Kerφ = H, then the cosets f…
automorphism
an isomorphism φ:G->G of a group with itself.
7 terms
Abstract Algebra
Group
Surjective function
Onto
Closure
Set with a binary operation defined that satisfies... Closure... As…
Onto; function f: X→Y... for all y in Y there exists an x in X s…
Surjective; function f: X→Y... for all y in Y there exists an x…
A set S is closed under operation f: X→X iff for all a,b in X…
Group
Set with a binary operation defined that satisfies... Closure... As…
Surjective function
Onto; function f: X→Y... for all y in Y there exists an x in X s…
34 terms
Abstract Algebra
Element
Empty Set
Well-Defined
Binary Operation
A set S is made up of elements, and if a is one of these elem…
There is exactly one set with no elements. It is the empty se…
A set is well-defined, meaning that if S is a set and a is so…
A binary operation * on a set S is a function mapping S x S i…
Element
A set S is made up of elements, and if a is one of these elem…
Empty Set
There is exactly one set with no elements. It is the empty se…
10 terms
Abstract Algebra
conditions for a group
dihedral group
elements a, b ∈ G are conjugate if...
conditions for an equivalence relation
binary operation, identity element, inverses, associativity
set of symmetries on a regular n-gon
there exists c ∈ G so that c⁻¹ * a * c = b.
reflexive, symmetric, transitive
conditions for a group
binary operation, identity element, inverses, associativity
dihedral group
set of symmetries on a regular n-gon
63 terms
Abstract algebra
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 surject…
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}
9 terms
Abstract Algebra
Binary Operation
Identity Element
Inverse Element
Group
A fcn *: G×G ⇨ G is called a binary operation on G (which mea…
There exists e ∈ G such that ... ea = ae... for every a ∈ G
For all a ∈ G there exists b ∈ G such that:... ab = ba = e... b…
A set G with a binary operation * is a group iff the followin…
Binary Operation
A fcn *: G×G ⇨ G is called a binary operation on G (which mea…
Identity Element
There exists e ∈ G such that ... ea = ae... for every a ∈ G
21 terms
Abstract Algebra
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 whene…
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 wh…
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 whene…
34 terms
Abstract Algebra
Element
Empty Set
Well-Defined
Binary Operation
A set S is made up of elements, and if a is one of these elem…
There is exactly one set with no elements. It is the empty se…
A set is well-defined, meaning that if S is a set and a is so…
A binary operation * on a set S is a function mapping S x S i…
Element
A set S is made up of elements, and if a is one of these elem…
Empty Set
There is exactly one set with no elements. It is the empty se…
30 terms
Abstract Algebra
Group
dihedral group
abelian/commutative group
order
set G with a binary operation '.': GxG→G that satisfies:... 1. (…
(Dn) group of symmetries of the regualr n-gon
if a₀b=b₀a for all a,b∈G
number of elements in the group... |Z4|=4
Group
set G with a binary operation '.': GxG→G that satisfies:... 1. (…
dihedral group
(Dn) group of symmetries of the regualr n-gon
Abstract Algebra
F
(a+b)+c=a+(b+c)... a+b=b+a... a+0=a... a+(-a)=0
It's an ideal I of a ring R that's gen…
An ideal is a subset of a ring, which…
What is a field?
What is a ring?
What is a principal ideal?
What is an ideal?
F
What is a field?
(a+b)+c=a+(b+c)... a+b=b+a... a+0=a... a+(-a)=0
What is a ring?
21 terms
Abstract Algebra
Definition: Congruence modulo m
What is the set of integers to which a…
Definition: Least non-negative residue
Proof: Least non-negative residue
Two integers a and b are congruent modulo m if if m divides a…
a ≡ b (mod m) for all b s.t. b = a + km, where k is some scal…
Let m be a natural number > 1. Every m ∈ N is congruent modul…
This can be proved via application of the division theorem. C…
Definition: Congruence modulo m
Two integers a and b are congruent modulo m if if m divides a…
What is the set of integers to which a…
a ≡ b (mod m) for all b s.t. b = a + km, where k is some scal…
Abstract Algebra Definitions
Binary Operation
Group
Order of a Group
Order of an Element
A function that assigns each ordered pair in G another elemen…
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 e…
Binary Operation
A function that assigns each ordered pair in G another elemen…
Group
A set G with binary operation. If associativity, inverses and…
39 terms
Abstract Algebra - Exam 3
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 E aH... AH = H iff a E H... AH = bH or aH (Union) bH = the empty…
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 r…
Cosets
G is a group, H c G... For all a in G, define aH = {ah | h E H}…
Poperties of cosets
a E aH... AH = H iff a E H... AH = bH or aH (Union) bH = the empty…
19 terms
Abstract Algebra Exam One
Generator
Cyclic Group
Centralizer
Subgroup
The element in the group whose order has the same number of e…
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 e…
Cyclic Group
Has a generator
25 terms
ABSTRACT ALGEBRA
isomorphism
homomorphism
epimorphism
monomorphism
同构
同态
满射
单射
isomorphism
同构
homomorphism
同态
12 terms
Abstract Algebra Exam 2
Let a, b ∈ Z such that a ≠ 0. Define a…
Define the order relation < on Z.
Let a, b ∈ Z such that a, b ≠ 0. Defin…
Let a, b ∈ Z and n ∈ N. Define a ≡ b.
Let a, b ∈ Z such that a ≠ 0. We say that a divided b, (or b…
Let a, b ∈ Z. We say that a is less than b, denoted a < b, if…
Let a, b ∈ Z such that a ≠ 0 and b ≠ 0. Then the greatest com…
Let n ∈ N. We say that an integer x is congruent to integer y…
Let a, b ∈ Z such that a ≠ 0. Define a…
Let a, b ∈ Z such that a ≠ 0. We say that a divided b, (or b…
Define the order relation < on Z.
Let a, b ∈ Z. We say that a is less than b, denoted a < b, if…
20 terms
Abstract Algebra Definitions Test 2
Subgroup
Centralizer
Center
Normalizer
Let G be a group. The subset H of G is a subgroup if H is non…
CG(A) = {g∈G | gag^-1=a for all a∈A}... CG(A) is the set of all…
Z(G) = {g∈G | gx=xg for all x∈G}... Z(G) is the set of elements…
NG(A) = {g∈G | gAg^-1 = A}... (G commutes with the set A But not…
Subgroup
Let G be a group. The subset H of G is a subgroup if H is non…
Centralizer
CG(A) = {g∈G | gag^-1=a for all a∈A}... CG(A) is the set of all…
13 terms
Abstract Algebra Definitions Test 3
Direct Product
Fundamental Theorem of finitely Genera…
Ring
Ring with identity
a direct product of Groups G and H is G x H = {(g,h) | g∈G, h…
Let G be a finitely generated abelian group, then G≅ℤ^r x ℤn1…
A ring R is a set together with 2 binary operations, + and *,…
a ring in which there is a multiplicative identity, 1R
Direct Product
a direct product of Groups G and H is G x H = {(g,h) | g∈G, h…
Fundamental Theorem of finitely Genera…
Let G be a finitely generated abelian group, then G≅ℤ^r x ℤn1…
44 terms
Abstract Algebra Definitions Test 1
Cardinality
Cartesian Product
Function
Domain
the number of elements in a set. also called order. |A| of a…
AxB of two sets A and B is an ordered pair (x,y) with x∈A, y∈…
f:A→B a function takes elements from A and maps them to B. *m…
the set A from a function
Cardinality
the number of elements in a set. also called order. |A| of a…
Cartesian Product
AxB of two sets A and B is an ordered pair (x,y) with x∈A, y∈…
27 terms
Abstract Algebra Test 2
a and b are equivalent mod n if
Six properties of equivalence classes
Define Group
Abelian group
n divides a - b
Communicative, associative, distrubutive, the identity exists…
a set paired with an operation (G,*) that satisfies associati…
a group that is communicative
a and b are equivalent mod n if
n divides a - b
Six properties of equivalence classes
Communicative, associative, distrubutive, the identity exists…
92 terms
Abstract Algebra Midterm 1
Division algorithm
The GCD of two non-zero integers a and…
If c divides a*b and gcd(c,a) = 1, then
Let p be a prime. If p|ab, then
for all integers a and b, there exist integers q and r such t…
the smallest positive integer that can be written as ax+by fo…
c divides b
either p|a or p|b (or both)
Division algorithm
for all integers a and b, there exist integers q and r such t…
The GCD of two non-zero integers a and…
the smallest positive integer that can be written as ax+by fo…
28 terms
Abstract Algebra test 1
Define set
The natural numbers (N)
The integers (Z)
The rational numbers (Q)
All x ∈ X satisfy P. Denoted X = {x: P}
{1,2,3,...}
{...,−1,0,1,2,...}
{p/q:p,q∈Z where q≠0}
Define set
All x ∈ X satisfy P. Denoted X = {x: P}
The natural numbers (N)
{1,2,3,...}
22 terms
Abstract Algebra "Math Symbols"
A'
V
set of elements not in A
"for all"
"or"
"and"
A'
set of elements not in A
"for all"
MTH 411 - Abstract Algebra
Well Ordering Principle
Division Algorithm
Greatest Common Divisor
a is congruent to b modulo n
Any nonempty set of positive integers contains a smallest mem…
Let a and b be integers with b>0. Then there exists unique in…
The ____ of two nonzero integers a and b is the largest of al…
Let a and b be integers and n a positive integer. When n divi…
Well Ordering Principle
Any nonempty set of positive integers contains a smallest mem…
Division Algorithm
Let a and b be integers with b>0. Then there exists unique in…
12 terms
Contemporary Abstract Algebra Definitions
Greatest Common Divisor
Least Common Multiple
Example of Modular Arithmetic
Equivalence Relation
The ... of two nonzero integers a and b is the largest of all…
The ... of two nonzero integers a and b is the smallest posit…
a=bn+r then r=a mod n
...on a set S is a set R of ordered pairs of elements of S su…
Greatest Common Divisor
The ... of two nonzero integers a and b is the largest of all…
Least Common Multiple
The ... of two nonzero integers a and b is the smallest posit…
Abstract Algebra Final
Ring
Ring with Unity
Commutative Ring
Zero-Divisor
A nonempty set R is called a ring if R is equipped with 2 bin…
If there is an element 1R∈R s.t. 1R×a=a×1R=a ∀a∈R and 1≠0... (id…
If a×b=b×a ∀a,b∈R
If R is a ring, then a∈R is a zero-divisor if there exists an…
Ring
A nonempty set R is called a ring if R is equipped with 2 bin…
Ring with Unity
If there is an element 1R∈R s.t. 1R×a=a×1R=a ∀a∈R and 1≠0... (id…
21 terms
Abstract algebra Theorems
Well-Ordered Principle
Lagrange
Fundamental Theorem of Finite Abelian…
Fundamental Theorem of Finitely Genera…
Every nonempty subset of the natural numbers is well ordered.
Let G be a finite group and let H be a subgroup of G. Then [G…
Every finite abelian group G is isomorphic to a direct produc…
Every finitely generated abelian group G is isomorphic to a d…
Well-Ordered Principle
Every nonempty subset of the natural numbers is well ordered.
Lagrange
Let G be a finite group and let H be a subgroup of G. Then [G…
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