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1/34
Binary operators
Groups
Subgroups, normal subgroups
Algebraic structures
NGUYEN CANH Nam1
1Faculty of Applied MathematicsDepartment of Applied Mathematics and Informatics
Hanoi University of Technologiesnamnc@mail.hut.edu.vn
HUT - 2010
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Binary operators
Groups
Subgroups, normal subgroups
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
Subgroups, normal subgroups
Definitions and examples
Properties of binary operators
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
Subgroups, normal subgroups
Definitions and examples
Properties of binary operators
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
Subgroups, normal subgroups
Definitions and examples
Properties of binary operators
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.
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Binary operators
Groups
Subgroups, normal subgroups
Definitions and examples
Properties of binary operators
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.
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Binary operators
Groups
Subgroups, normal subgroups
Definitions and examples
Properties of binary operators
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).
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Binary operators
Groups
Subgroups, normal subgroups
Definitions and examples
Properties of binary operators
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
Subgroups, normal subgroups
Definitions and examples
Properties of binary operators
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
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Binary operators
Groups
Subgroups, normal subgroups
Definitions and examples
Properties of binary operators
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
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Binary operators
Groups
Subgroups, normal subgroups
Definitions and examples
Properties of binary operators
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
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Binary operators
Groups
Subgroups, normal subgroups
Definitions and examples
Properties of binary operators
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
.
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y p
Groups
Subgroups, normal subgroups
Definitions and examples
Properties of binary operators
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.
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y p
Groups
Subgroups, normal subgroups
g p
Concepts on groups
Basic properties of 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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Groups
Subgroups, normal subgroups
Concepts on groups
Basic properties of 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
Subgroups, normal subgroups
Concepts on groups
Basic properties of 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
Subgroups, normal subgroups
Concepts on groups
Basic properties of 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.
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G
Semigroups
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Groups
Subgroups, normal subgroups
Concepts on groups
Basic properties of 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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Groups
Semigroups
Concepts on groups
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Groups
Subgroups, normal subgroups
Concepts on groups
Basic properties of 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.
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Groups
Subgroups, normal subgroups
Concepts on groups
Basic properties of 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.
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Groups
Subgroups, normal subgroups
Concepts on groups
Basic properties of 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.
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Groups
Subgroups, normal subgroups
Concepts on groups
Basic properties of 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.
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Groups
Subgroups, normal subgroups
Concepts on groups
Basic properties of 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 operatorsGroups
SemigroupsConcepts on groups
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p
Subgroups, normal subgroups
p g p
Basic properties of groups
GroupsProperties
PropositionIn a group, the identity element is unique; for an element x its
inverse element is unique
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SemigroupsConcepts on groups
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p
Subgroups, normal subgroups
p g p
Basic properties of groups
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.
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Subgroups, normal subgroupsNormal subgroups
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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Subgroups
Normal subgroups
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Subgroups, normal subgroupsNormal subgroups
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}.
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S b l b
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Normal subgroups
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Subgroups, normal subgroupsNormal subgroups
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.
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S bg l bg
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Subgroups, normal subgroupsg p
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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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
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Subgroups normal subgroups
Subgroups
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Subgroups, normal subgroups
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.
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Subgroups normal subgroups
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Subgroups, normal subgroups
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.
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Subgroups, normal subgroups
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Subgroups, normal subgroups
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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