chapter v - algebraic structures i

8/4/2019 Chapter v - Algebraic Structures I

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Binary operators

Groups

Subgroups, normal subgroups

Algebraic structures

NGUYEN CANH Nam1

1Faculty of Applied MathematicsDepartment of Applied Mathematics and Informatics

Hanoi University of [email protected]

HUT - 2010

NGUYEN CANH Nam Mathematics I - Chapter 5
http://goforward/http://find/http://goback/


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Binary operators

Groups


Agenda

1 Binary operators

Definitions and examples

Properties of binary operators

2 GroupsSemigroups

Concepts on groups

Basic properties of groups

3 Subgroups, normal subgroupsSubgroups

Normal subgroups



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Binary operators

Groups




Agenda

1 Binary operators



2 GroupsSemigroups

Concepts on groups



Normal subgroups



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Binary operators

Groups




Binary operatorsDefinition

Definition

A mapping

: S S Sis called a binary operator on the set S.

A binary operator on S thus assigns to each ordered pair of

elements of S exactly one element of S. Binary operators are

usually represented by symbols like , , +, , instead of lettersf, g and so on. Moreover, the image of (x, y) under a binaryoperator is written x y instead of (x, y).



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Binary operators

Groups




Binary operatorsExamples

Example

Addition (+) and multiplication () in the set ZZ of integers,more generally, in the set IR of real numbers are the most

familiar examples of binary operators.If A and B are subset of X, then A B, A B, and A Bare also subsets of X. Hence, union, intersection and

difference are all binary operators on the set P(X).

Again, given mappings f : X X and g : X X, theircomposition f g is also a mapping X X. Hence, thecomposition () of mappings is a binary operator on the setS = XX of all mappings from X to X.


Bi


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Binary operators

Groups




Binary operatorsClosure

Definition

Let be an operation on a set B and C B. The subset C issaid to be closed under the operation provided that

Whenever a, b

C then a

b

C

Example

Let ZZ be the set of integer numbers, O be the subset containing allodd numbers and E be the set containing all even numbers. Then

a) Under ordinary addition E is closed.

Actually, if a, b E (even) then a+ b E (even).b) The set O is not closed under ordinary addition.

Indeed, if a, b

O (odd) then a+ b is even, doesnt belong to O.


Bi t


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Binary operators

Groups




Binary operatorsClosure

Example

Let OQ be the set of rational numbers and define

B = {a+ b

2 | a, b OQ}. Then B is closed under ordinaryaddition and multiplication on OQ.

Actually,a1 + b1

2 + a2 + b2

2 = (a1 + a2) + (b1 + b2)

2

and(a1 + b1

2)(a2 + b2

2) = (a1a2 + 2b1b2) + (a1b2 + a2b1)

2

Example

Given X Y, the power set P(X) is closed under the operationof union and intersection on the power set P(Y).


Binary operators


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Binary operators

Groups




Agenda

1 Binary operators



2 GroupsSemigroups

Concepts on groups



Normal subgroups


Binary operators


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Binary operators

Groups




Binary operatorsCommutativity and Associativity

Definition

A binary operation : S S S on the set S is1 commutative if

x

y = y

x for all x, y

S

Addition (+) and multiplication () in the sets IN,ZZ and IR arecommutative :

a+ b = b+ a, a b = b a2 associative if

x (y z) = (x y) z for all x, y, z SAddition (+) and multiplication () in the sets IN,ZZ and IR areassociative :

a+ (b+ c) = (a+ b) + c, a

(b

c) = (a

b)

c


Binary operators


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Binary operators

Groups




Binary operatorsIdentity element

Definition

Identity element. We say that a binary operation : S S Son the set S has the identity element e if e

S and

a e = e a= a for all a S

Example

Addition (+) in the sets IN,ZZ and IR has the identity element 0 :

a+ 0 = 0 + a= aMultiplication () in the sets IN,ZZ and IR has the identity element1 : a 1 = 1 a= a


Binary operators


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Binary operators

Groups




Binary operatorsInverse element

Definition

Inverse element. Let : S S S be an binary operation on the set S withthe identity element e. Consider an element a S, element a S is called

the inverse of a if a a = a a= e

If every element of S has the inverse element we say that operator is

invertible.

Example

Addition (+) in the sets ZZ and IR, all element ahas the inverse elementa : a+ (a) = (a) + a= 0Multiplication () in the sets ZZ, all element a= 1 has no inverse element.Multiplication () in the sets IR, all element a= 0 has the inverse element 1

a,

we write a1, a1

a=

1

a a= 1


Binary operatorsD fi i i d l


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Binary operators

Groups




Binary operators

Remark

a) If a binary operator denoted by + then the identity element

is usually denoted by0 and the inverse element of x isusually denoted byx called the negative of x.

b) If a binary operator denoted by then the identity elementis usually denoted by1 and the inverse element of x is

usually denoted by x1

.


Binary operatorsD fi iti d l


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y p

Groups




Binary operators

Example

Let Sym(X) be the set of all bijections from on X. Thecomposition operation

has the following properties

a) (f g) h = f (g h) (associative),b) The identity element is idX,

c) For f Sym(X), its inverse element the inverse map f1.

d) The operation is not commutative since, as you mightremember, in general, f g= g f.


Binary operators Semigroups


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y p

Groups


g p

Concepts on groups


Agenda

1 Binary operators



2 GroupsSemigroups

Concepts on groups



Normal subgroups




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Groups


Concepts on groups


SemigroupsDefinitions

The simplest algebraic structure to recognize is a semigroups

which is defined as follows.

Definition

Let S be a non-empty set on which there is defined a binaryoperation denoted by . For a, b S the outcome of theoperation between aand b is denoted by a b. Then set S iscalled a semigroup if the following axioms hold

(i) For all a, b

S, a

b

S (closure).

(ii) For all a, b c S, (a b) c = a (b c) (associativity).

Any algebraic structure S with a binary operation + or isnormally written (S, +) or (S, )




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Groups


Concepts on groups


SemigroupsExamples

Example

(a) The system of integers or reals under usual multiplication

(or addition)

(b) The set of mappings from a nonempty set S into itself

under composition of mappings.




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Groups


Concepts on groups


SemigroupsExamples

Semigroups are common in mathematics and easy to create as

the following examples show

Example

1 Let S be the set of one element, S= {

a}

, say. Define

a a= aThen (S, ) becomes a semigroup.2 Let S be a non-empty set. Define a binary operation on

S by a b = b for all a, b S.We clain that S is a semigroup under this operation.

Certainly, we have for ll a, b S, a b = b S. We alsohave for all a, b, c S, (a b) c = b c = c anda (b c) = a c = c and so (a b) c = a (b c). Thus(S, ) is a semigroup.


Binary operators

G

Semigroups

C t


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Groups


Concepts on groups


Agenda

1 Binary operators



2 GroupsSemigroups

Concepts on groups



Normal subgroups


Binary operators

Groups

Semigroups

Concepts on groups


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Groups


Concepts on groups


GroupsDefinition

Definition

A group is a nonempty set G equipped with an binary operation

that satisfies the following axioms(i) Closure : If a

G and b

G, then a

b

G.

(ii) Associativity : a (b c) = (a b) c for all a, b, c G.(iii) There is an element e G (called the identity element)

such that a e = a= e a for every a G.(iv) For each a

G, there is an element a

G (called the

inverse of a) such that a a = a a= e.If the binary operation in a group (G, ) is commutative - that is,a b = b a for all a, b G - then the group is calledcommutative or abelian.


Binary operators

Groups

Semigroups

Concepts on groups


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Groups


Concepts on groups


GroupsExamples

Example

1 Let ZZ be the set of all integers and let be the ordinaryaddition, +, in ZZ. That ZZ is closed and associative under are basic properties of the integers. What serves as the

unit element, e, of ZZ under ? Clearly, sincea= a e = a+ e, we have e = 0, and 0 is the requiredidentity element under addition. What about a1 Here too,

since e = 0 = a a1 = a+ a1, the a1 in this instance isa, and clearly a (a) = a+ (a) = 0. Moreover, it is aAbelian group.However, ZZ under multiplication is not a group. We know

that, for example, the number 2 does not have an inverse

element in ZZ.


Binary operators

Groups

Semigroups

Concepts on groups


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Groups


Concepts on groups


GroupsExamples

Example

2 Let OQ be the set of all rational numbers and let the

operation on OQ be the ordinary addition of rationalnumbers. As above, OQ is easily shown to be a group under

. Note that ZZ OQ and both ZZ and OQ are groups under thesame operation . They are all Abelian group.

3 Let IR+ be the set of all positive real numbers and let the

operation on IR+ be the ordinary product of real numbers.Again it is easy to check that IR

+

is an Abelian group under.4 Sym(X) with the composition of maps is a

noncommutative group.


Binary operators

Groups

Semigroups

Concepts on groups


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Groups


Concepts on groups


GroupsOrder

A group G is said to be finite if it has a finite number of

elements, otherwise it is said to be infinite. If G is finite , the

number of elements in G is called the order of G and is denoted|G|. An infinite group is often said to have infinite order.Example

ZZ and OQ under multiplication are examples of groups of infinite

order.


Binary operatorsGroups

SemigroupsConcepts on groups


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Groups


Concepts on groups


Agenda

1 Binary operators



2 GroupsSemigroups

Concepts on groups



Normal subgroups





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p


p g p


GroupsProperties

PropositionIn a group, the identity element is unique; for an element x its

inverse element is unique





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p


p g p


GroupsProperties

Theorem

Let(G, ) be a group.(i) (Cancellation) Let a, b, x G be such that a x = b x. Then

a= b. (Similary y a= y b, y G implies a= b.)(ii) (Unique solution of equation) Let a, b G. Then the equation

a x = b has the unique solution x = a1 b. (Similary y a= bhas the unique solution y = b a1.

Proof. (ii) Certainly a

1

b is a solution sincea (a1 b) = (a a1) b = e b = b where e is the identity.On the other hand, a x = b implies that aa (a x) = a1 b fromwhich we conclude that (a1 a) x = a1b and sox = e x = a1 b.





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Subgroups, normal subgroups Basic properties of groups

GroupsProperties

TheoremLet(G, ) be a group and a, b G. Then

(a b)1 = b1 a1.



Subgroups

Normal subgroups


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Subgroups, normal subgroupsNormal subgroups

Agenda

1 Binary operators



2 GroupsSemigroups

Concepts on groups



Normal subgroups



Subgroups

Normal subgroups


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SubgroupsDefinition

Definition

Let H be a non-empty subset of the group (G, ) which is also agroup under the operation . Then H is called a subgroup of G.

Example

1 Two obvious subgroups of G are G itself and {e}.2 ZZ is a subgroup of (OQ, +) or (IR, +).

3 2ZZ = {2n | n ZZ} is a subgroup of (ZZ, +).4 H = {2n+ 1 | n ZZ} is not a subgroup of (ZZ, +).5 ZZ is not a subgroup of (OQ, ) where OQ = OQ \ {0}.



S b l b

Subgroups

Normal subgroups


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SubgroupsCriterion conditions

Theorem (Subgroup criterion)

A nonempty subset H of the group(G, ) is a subgroup of G ifand only if two following conditions hold

(i) For all a, b H we have a b H.(ii) For all a H we have a1 H.

Proposition

The intersection of a collection of subgroups of G is also a

subgroup.



S bg l bg

Subgroups

Normal subgroups


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Subgroups, normal subgroupsg p

Agenda

1 Binary operators



2 GroupsSemigroups

Concepts on groups



Normal subgroups



Subgroups normal subgroups

Subgroups

Normal subgroups


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Subgroups, normal subgroupsg p

CosetsDefinition

Definition

Let H be a subgroup of a group (G, ). The left and the right cosets ofH containing g are

gH = {g h | h H} and Hg = {h g | h H}respectively.

Proposition

Assume that H is a subgroup of the group (G, ). Then

(i) For x G, x xH.(ii) If y xH then xH = yH.

(iii) The cosets of H form a partition of G.

(iv) xH = yH if and only if x1 y H.NGUYEN CANH Nam Mathematics I - Chapter 5



Subgroups

Normal subgroups


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Normal subgroupsDefinition

Definition

A subgroup H of a group G is called a normal subgroup of

(G,

) if for all x

G, h

H we have x

h

x1

H. In this

case we write H 0 G.

Example

1 Let G be a group and e is the identity element of G. Then

G and {e} are normal subgroups of G.2 If G is an Abelian group then any subgroup H of G is a

normal subgroup.




Subgroups

Normal subgroups


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Quotient groupDefinition

Let H be a normal subgroup of (G, ). Put x = xH for x G.On the set G/H = {xH | x G} = {x | x G} we define anoperator as follows

x y = x y.Then this is a binary operator.

Proposition

Assume that H is a normal subgroup of G. G/H with the aboveoperator id also a group, called the quotient group.




Subgroups

Normal subgroups


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Quotient groupExample

Example

Let ZZ be the set if integers, m be a fix natural number,mZZ = {mn | n ZZ}. Then mZZ is a normal subgroup of theadditive group ZZ and the quotient group

ZZ/mZZ = {0, 1, , m 1} where k = {mn+ k | n ZZ}.


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