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MATHEMATICS

UNIT-III

ALGEBRA GROUP THEORY:

Non empty set G together with a binary operation * is called a group.

If its satisfy the following condition.

(i) Closure: ∀ a,b ɛ G ⇒a*b ɛ G

(ii) Associative: ∀ a,b,c ɛ G ⇒ a* (b *c) = (a*b)*c

(iii) Identity: There is an elements e ɛ G such that a*e=e*a=a ∀ a ɛ G

(iv) Inverse: ∀ a ɛ G There exist an elements a-1

ɛ G Such that a*a-1

=a-1*

a=e

a-1

is called inverse of a.

Ex: Z, Q, R and C are groups under usual addition.

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ABELIAN GROUP:

A Group (G,*) is said to be a abelian. If its satisfies the commutative

property a*b=b*a ∀ a ,b ɛ G.

GROUPOID:

A set G with binary composition is said to be a groupoid.

SEMI GROUP:

A set G with a binary composition which is associative is said to be

semigroup.

MONOID:

A set G with a binary composition which is associate and identity element

exist is said to be monoid.

ORDER OF G:

The number of elements of group G is called an order of G and is denoted

by O(G) .If O(G) is finite. The group G is said to be finite.

PROPERTIES OF GROUP:

If G is a group then G is a group.

(i) Identity element of G is unique.

(ii) Every element in G, has a unique inverse.

(iii) ∀ a ɛ G, (a-1

)-1

=a.



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(iv) (a * b)-1

=b-1*

a-1

,for all a , b ɛ G

(v) a,x,y ɛ G and a*x=a*y then x=y (Left cancellation law)

(vi) a,x,y ɛ G and x*a=y*a then x=y (Right cancellation law)

The equation a*x=b and y*a=b have unique solution.

ONE TO ONE:

The mapping f:G→Gˡ is said to be one to one .

f(x) = f(y) ⇒ x=y (or)

x≠y=>f(x)≠f(y)

⇒f(x)≠f(y)

PERMUTATION:

A one to one and onto mapping of a finite set, onto itself is called

permutation, and then this permutation replaces n objects cyclically it is called

cyclic permutation.

HOMOMORPHISM:

A mapping φ from a group G, into a group G is said to be a homomorphism

if a,b ɛ G. φ(ab)=φ(a)φ(b).

ISOMORPHISM:

If G→Gˡ such that f is one to one and onto mapping preserving group

composition in G and Gˡ.F(a*b) = f(a)*f(b) then G is isomorphism to Gˡ

There fore G⪮Gˡ



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PROPERTIES OF ISOMORPHIC MAPPING:

Identities and inverse correspond f(e) is identity of Gˡ .where e is identity

of G. There fore, f(a-1

)=[(a )]-1

.

There is image of inverse of an element is inverse of image of that

element.

SUBGROUP:

A non empty subset H of a group G is subgroup of G iff

(і). ∀ a, b ɛ H a* b ɛ H (ii).∀ a ɛ H a-1

ɛ H

A non empty subset H of a group G, is a subgroup of G iff ∀

a,b ɛ H ⇒ a*b-1

ɛ H

Let H be a finite subset of group G, H is a subgroup of G iff ∀ a, b ɛ H

⇒ a*b ɛ H.

THEOREM: If H, K are a subgroup of G, HK is a subgroup of G HK=KH.

COROLLARY:

If H and K are a subgroup of abelian group G then HK is a subgroup of G.

THEOREM:

If H and K finite subgroup of G then o(HK) =

.



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SSRRIIMMAAAANN NORMAL SUBGROUP:

If N is normal subgroup of group G if g ɛ G and n ɛ N gng-1

ɛ N.

THEOREM:

A subgroup N of a group G is a normal subgroup of G iff gNg-1

=N g ɛ G

THEOREM:

If N is a normal subgroup of group G iff every left coset N in G is a right

coset of N in G.

THEOREM:

If N is a normal subgroup of group G iff the product two right coset of G is

also a right coset of G.

QUOTIENT GROUP:

Let N be a normal subgroup of G and G/N is set of all is right coset of G.

Then quotient group defined as G/N = {Na/ a ɛ G}.

THEOREM:

If G is a group N is a normal subgroup of G. Then G/N is also a group.

THEOREM:

If G is a finite group and N is a normal subgroup of G then

(G/N) = O(G)/O(N).



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LEMMA:

Suppose G is a group, N is a normal subgroup of G. Define the mapping

φ: G →G/N by φ(x) = N x, x ɛ G then φ is a homomorphism of G onto G/N.

KERNAL OF A Φ:

If φ is a homomorphism of G to G that erne of φ, Kφ is defined by.

Kφ = {x ɛ G/φ(x) = e , e is the identity e ement of G }.

LEMMA:

If φ is a homomorphism of G into G then (і) φ(e)= e is the identity

e ement of G . (ii) φ(x-1)=(φ(x))

-1 ∀ x ɛ G.

LEMMA:

If φ is homomorphism of G into G , the kernel K is a normal subgroup of

G.

ISOMORPHISM:

A homomorphism of from G into G is said to be an isomorphism, if φ is

one to one.

FUNDAMENTAL THEOREM OF HOMOMORPHISM:

Let φ be a homomorphism of G into G ith erne K.Then G/K isomorphic

G .



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CAUCHY’S THEOREM FOR ABELIAN:

Suppose G is a finite abelian group p/O(G) where P is a prime number then

there is an e ement a ≠ e ɛ G such that aP=e.

SYLOW’S THEREM FOR ABELIAN:

If G is finite abelian group and if p is prime such that pα∤o(G) and p

α+1∤o(G)

then G has subgroup order pα.

CONJUGATE CLASS:

Let a Є G and Ca={x ɛ G/ a ∽ λ}.Ca consists of the set of all distinct elements

y-1

ay as y ranges over G.

NORMALIZER:

If a ɛ G and N(a) is said to be normalizer then N(a)={x ɛ G/ax=xa}

(ie.,) N(a) consists of those elements in G Which commute with a.

LEMMA:

N(a) is a subgroup of G.

CENTER OF G:

The center Z(G) of a group G is defined as Z(G)={a ɛ G/ xa = ax ∀ x ɛ G}.

THEOREM: (і) If a ɛ Z if and only if N(a) = G

(ii) If G is finite a ɛ Z if and only if O(N(a)) = O(G)

THEOREM:

If O(G)=P2 ,where p is prime then prove that G is abelian.



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We know the order of a subgroup of a finite group G must divide |G|. If

G is a abelian then there exist subgroups of every order dividing |G|. This is not

true for non abelien groups; A4 can be shown to have no subgroup of order 6.

The Sy o ’s theorems assert for prime po er dividing |G|, there is a

subgroup of that prime-power order .They also give some information about

the number of such subgroups.

THEOREM: (Sylow’s theorem second part)

Let P1 and P2 be sylow p-subgroup of a finite group G. Then P1 and P2 are

conjugate subgroups of G.

THEOREM: (Sylow’s theorem third part)

If G is a finite group and p divides |G|, then the number of Sylow p-subgroup

is 1+kp.

EXTENSION FIELD:

K is an extension field of F, then it is denoted by [K:F]. Also called as K is

a vector space over F.

(ie) [K:F] is a finite then it is called K is a finite extension of F.

THEOREM:

If L is a finite extension of K and if K is a finite extension of F, then L is a

finite extension of F. (ie) [L:F]=[L:K][K:F].



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SSRRIIMMAAAANN ALGEBRAIC:

Let K be an extension field of F, then an element a ϵ K is said to be algebraic

over F there exists e ement α0, α1, α2, … , αn in F not all zero such that

α0+α1a1+α2a

2+…+αna

n = 0.(ie) a ɛ K is algebraic over F if there exists a non-

zero polynomial p(x) ɛ F[x] such that p(a)=0

THEOREM:

An element a ɛ K is algebraic over F if and only if F(a) is a finite extension

of F (or) [F(a):F] = n.

ALGEBRAIC NUMBER:

A complex number is said to be an algebraic number if it is algebraic over the

field of rational numbers.

A complex number which is not algebraic is called transcendental.

ROOTS:

If p(x) ɛ F[x], an element 'a' in some extension field K of F is called a root of

p(x) if p(a) = 0.

REMAINDER THEOREM:

If p(x) ɛ F[x] and if K is an extension of F, then for any element b ɛ K,

p(x) = (x-b)q(x) + p(b) where q(x) ɛ K[x] and where deg q(x) = deg p(x) -1

ROOT OF MULTIPLICITY M:

An element a ϵ K is a root of p(x) ɛ F[x] of multiplicity m if (x-a)m/p(x) but

(x-a)m+1⫮ p(x).



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SSRRIIMMAAAANN THEOREM :

A polynomial of degree n over a field can have n-roots in any

extension field.

SPLITTING FIELD:

Let f(x) ɛ F[x]. A minimal extension field E over F containing all

roots of f(x) is called splitting field of f(x), that is no smaller field extension other

then E not to contain all roots of f(x).

THEOREM:

If f(x),g(x) ɛ F[x], α ɛ F then

(1). [f(x)+g(x)]' = f '(x) + g '(x)

(2). [α f(x)]' = α f '(x)

(3). [f(x)g(x)]' = f '(x) g(x) + f(x) g'(x)

THEOREM:

A polynomial f(x) ɛ F[x] has a multiple root if and only if f(x) and f '(x)

have a non-trivial common factor over F.

CHARACTERISTIC OF F(FIELD):

A fie d F is said to be characteristic m if ma ǂ 0 for a ǂ 0 in F and m > 0.

REMARK:

Char F=0 => the field F has infinite number of elements if there exists

a ɛ K such that K = F(α).

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