Discrete Mathematics by Meri Dedania Assistant Professor MCA department

Prepared By Meri Dedania (AITS)

Discrete Mathematicsby

Meri DedaniaAssistant ProfessorMCA department

Atmiya Institute of Technology & ScienceYogidham Gurukul

Rajkot


Group Theory


Definition of Group A Group < G , > is an algebraic system

in which on G satisfies four condition Closure Property For all x , y G x y G Associative Property For all x , y , z G x (y z) = (x y) z Existence of Identity element There exists an element e G such that for any a

G x e = x = e x Existence of Inverse Element For every x G ,there exists an element denoted by

a-1 G such that x-1 x = x x-1 = e


Definition of abelian Group A Group < G , >in which the

operation is commutative is called abelian Group i.e. a,b G , a b = b a

Example1. < Z , + > is Abelian Group2. < Q , + > is abelian Group


Theorem 1 : Let e be an identity element in group < G , > , Then e is unique

Proof : Let e and e` are two identity in G e e` = e if e` is identity e e` = e` if e is identity since ee` is unique element in G e = e`

Properties of Group


Theorem 2 : Inverse of each element of a group < G , > is unique

Proof : Let a be any element of G and e the

identity of GSuppose b and c are two different inverse of

a in G. a b = e = b a (if b is an inverse of a) a c = e = c a (if c is an inverse of a) Now , b = b e = b ( a c) = (b a) c = e c = cThus a has unique inverse


Theorem 3 : if a-1 is the inverse of an element a of group < G , > then (a-1)-1=a

Proof : Let e be the identity of Group < G

, > a-1 a = e (a-1)-1 (a-1 a) = (a-1)-1 e ((a-1)-1 a-1) a = (a-1)-1

e a = (a-1)-1

(a-1)-1 = a


Theorem 4 : If < G , > be a group then for any two elements a and b of < G , > prove that ( a b )-1 = b-1 a-1 rule of reversal

Proof : Let a-1 and b-1 are inverse of a and b respectively

and e be the identity a a-1 = e = a-1 a b b-1 = e = b-1 b (a b) (b-1 a-1) = [(a b) b-1] a-1 = [a (b b-1)] a-1 = [a e] a-1 = a a-1

= e Similarly , (b-1 a-1) (a b) = e This show that b-1 and a-1 is inverse of b and a Hence , ( a b )-1 = b-1 a-1


Cancellation Property : if a , b and cbe any three elements of a group < G , > then

ab = ac b = c left cancellation ba = ca b = c right cancellationProof : Let a G and also a-1 G aa-1 = e = a-1a where e is identity of G Now , ab = ac a-1(ab) = a-1 (ac) (a-1 a) b = (a-1 a)c e . b = e . c b =c similarly , ba = ca b = c


• Definition of Permutation A permutation is one to one mapping of non empty set P , say onto itself Example : Let S = {1,2,3} Then function f : S S f(1) = 2 f(2) = 3 f(3) = 1 Then permutation P1 = P2 =

Permutation Group


P3 = P4 =

P5 = P6 =

There are n! of pattern of expressing Permutation .

So if Set has 3 elements then pattern of expressing permutation is 3! = 6


Equality of Permutations : Let f and g be two permutations

defined on a non empty set P. Then f = g if and only if f(x) = g(x) x P

Example 1) Let S = {1,2,3,4} and let

permutation f and g are equal or not..

f = g = 2) Let S = {1,2,3,4} and let

permutation f and g are equal or not..

f = g =


Permutation Identity An Identity permutation on S ,

denoted by I , is defined as I(a) = a a S

For example :

f =

Note : In identity permutation the image of element is element itself


Composition of Permutation ( Product of Permutation)

Let f and g be two arbitrary permutations of like degree , given by,

f = g = on non empty set A. Then the

composition (or Product) of f and g is defined as

Continue…


f g = = Example Let P1 = P2= P3 =

Check P1 (P2 P3) = (P1 P2) P3


• Inverse Permutation Every permutation f on set P =

{a1,a2,a3,…,an} Possesses a unique inverse permutation , denoted by f-1 thus if

f = Then f-1 =


Cyclic PermutationLet t1,t2,…..,tr be r distinct

elements of the set P = {t1,t2,…., tn}.Then the permutation p : P P is defined by

p(t1) = t2 , p(t2) = t3,….,p(tr-1)= tr, p(tr)=t1 is called a cyclic permutation of length r.

Example : The permutation P = is written as

(1,2) , (3,4,6) , (5).. The cycle (1,2) has length 2 , The cycle length 3,The cycle 1.


Definition of Cyclic Group If there exists an element a G

for some group < G , > such that every element of G can be written as some power of a , that is an for some integer n. then a Group

< G , > is said to be cyclic Group

Every Cyclic Group is abelian

Example for set A = { , , ,} and binary operation


I. If < G , > is an abelian group , then for all a , b G show that ( a b )n = an bn

Solution ( a b )n = an bn

( a b )n+1=an+1 bn+1

( a b )n+2=an+2 bn+2

Now , ( an bn ) ( a b ) = ( a b )n+1

= ( an+1 bn+1 ) (bn a )=(a bn)

By cancellation , similarly bn+1 a = a bn+1

Again bn+1 a = b(bn a) = b(abn) i.e., abn+1 = b(abn )


III. Show that in a Group < G , > , if for any a, b G , ( a b )2 = a2 b2, then <G , > must be abelian

Solution : Let < G , > be a Group and let a , b

G ( a b )2 = a2 b2

( a b ) ( a b ) = ( a a) ( b b ) a ( b a) b = a ( a b) b By left and right cancellation property b a = a b Thus we have a b = b a . a,b G Hence < G , >is an abelian Group


IV. Show that if every element in a group is its own inverse , then the group must be abelian

Solution : Let a , b G a b G (by closure property)Now, a-1 = a and b-1 = b ( a b)-1 = a bNow, ( a b)-1 = a b b-1 a-1 = a b b a = a b Thus we have a b = b a , a,b G Hence < G , >is an abelian Group


V. Write down Composition table for <Z7, +5> and <Z7

*, 7> where Z7

* = Z7 - {0}

+7

0 1 2 3 4 5 6

0 0 1 2 3 4 5 61 1 2 3 4 5 6 02 2 3 4 5 6 0 13 3 4 5 6 0 1 24 4 5 6 0 1 2 35 5 6 0 1 2 3 46 6 0 1 2 3 4 5

7

1 2 3 4 5 6

1 1 2 3 4 5 62 2 4 6 1 3 53 3 6 2 5 1 44 4 1 5 2 6 35 5 3 1 6 4 26 6 5 4 3 2 1


VII. Show that < {1} , > and < {1 , -1} , > are the only finite groups of nonzero real numbers under the operation of multiplication

Solution:


• Definition of Sub Group : Let < G , > be a Group and S G, such

that it satisfies the following condition:1) e G , Where e is the identity of < G , > 2) For any a S , a-1 S3) For a , b S , a b SThen < S , > is called Sub Group of <G , >

For any group <G , > , <{e} , > and <G , > are Trivial Sub Groups of <G , >.Let <Z-{0}, X> is a Group then <{1}, X> & <Z,X> are Trivial Sub group of <Z, X>All other subgroups of <G , > are called Proper Subgroup Let < {,-1,0,1} , X > is Proper subgroup of <Z , X >

Sub Group and Homomorphism


Theorem : A subset S of G is a subgroup of < G , > iff for any pair of elements a , b S , a b-1 S

Proof : Assume that S is a subgroup if a , b S then b-1 S and a b-1 S To prove the converse , let us assume that

a , b S and a b-1 S for any pair a , b. taking b = a , a a-1 = e SFrom e , a, b S e a-1 = a-1 SSimilarly , b-1 S.Finally , because a and b-1 are in S , we have

a b S.Hence , < S , > is a sub group of < G , >


Definition of Group Homomorphism

Let < G , > and < H , > be two Group. A mapping g : G H is called a group homomorphism from < G , > to < H , > if for any a , b G

g (a b) = g(a) g(b) g(eG) = eH

g(a-1) = [g(a)]-1


Definition of Group Isomorphism

Let f : < G , > < H , >.if f is one to one and onto. Then Group is called isomorphism

A homomorphism f : < G , > < H , > is called an endomorphism

A Isomorphism f : < G , > < H , > is called an automorphism


Definition Kernal of Homomorphism

Let < G , > and < H , > be two Groups and let f is homomorphism of G into H. The set of elements of G which are mapped into eH , the identity of H is called the kernal of the homomorphism and is denoted by Kf or Ker(f)


Theorem : The Kernal of homomorphism f : <G , > < H , > is sub group of < G , >

Proof : Here f : < G , > < H , > is homomorphism Ker (f) = {x G | f(x) = eH identity element of H} k (f) because eG K(f) (f(eG)=eH) let a , b Kf f (a) = eH & f(b) = eG Now, f(ab-1) = f(a) . f(b-1) = f(a) . [f(b)]-1

= eH . eH-1

= eH . eH = eH ab-1 Kf Kf is a sub group of < G , >


• Show that every interval of lattice is a sub lattice of a lattice

Proof: Let < L , > be a lattice and a , b L a , b a a b a [ a , b ] also [a , b] = {x L | a x b} L Let x , y [ a , b ] a x b , a y b a a x y b b a a x y b b a x y b a x y b x y [ a , b ] and x y [ a , b ] [ a , b ] is sub lattice of the lattice < L , >


Draw Hasse Diagram of the poset {2,3,5,6,9,15,24,45},D . Find

(i) Maximal and Minimal elements(ii) Greatest and Least members, if exist.(iii) Upper bound of {9,15} and l.u.b. of{9,15} , if exist.(iv) Lower bound of {15,24} and g.l.b. of{15,24} , if exist.


Definition Of Right Cosets Let G be a Group and H is any sub

Group of G. Let a be any element of G . Then set Ha = {ha : hH} is called a right coset of H in G generated by a.• Definition of Left Cosets Let G be a Group and H is any sub

Group of G. Let a be any element of G . Then set aH = {aH : hH} is called a right coset of H in G generated by a.


Lagrange’s Theorem : The order of each sub group of a

finite group G is a divisor of the order of G

Index in G : The number of left cosets of H in G

is called index of H in G. Definition of Normal

subgroup: A sub group < H , > is sub

group of < G , > is called a normal sub group if for any a G , aH = Ha


Fins the Sub group of < Z12 , +12 > show that <{1,4,13,16} , 17> is

subgroup of < Z17* , 17 >

Show that every sub group of abelian group is normal

Let x G and h HXhx-1 = xx-1h = eh = h x G and h H xhx-1 H i.e. xH = Hx H is normal subgroup of G


Definition of Irreflexive A relation R on a set A is

irreflexive if aRa for a A, if (a,a) R

For example A = {1,2,3,4}R =

{<1,1>,<1,2>,<2,3>,<1,3>,<4,4>}

R = {<1,3>,<2,1>}

Discrete Mathematics by Meri Dedania Assistant Professor MCA department

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