GROUPS......

Binary operation on a set:

Let S be a nonempty set then an operation * on a set S is said to be a binary operation on S if a * b belongs to S for every a,b in S.

ie: * from S x S to S defined by * ( a,b ) = a * b for every ( a, b ) in S x S.

Eg: Let R be the set of all real numbers then +, -, x are binary operations on R . But division is not a binary operation on R because division of any real number by zero is not defined.

Algebraic structure:

A non- empty set G together with one or more binary operations is called an algebraic structure. It is denoted by , etc.

Eg: ( G , * ),( G , *, o )

Group:

An algebraic structure ( G , * ) is said to be a group if the following conditions are satisfied,

G1: * is associative in G

ie: a * ( b * c ) = ( a * b ) * c for every a,b,c in G

G2: There is a unique element e in G such that a * e = a = e * a for every ‘ a ‘ in G.

(Existence of identity element)

G3: For every element ‘ a ‘ in G there is an element ‘ a¹ ‘ in G such that

a * a¹ = e = a¹ * a. (Existence of inverse element)

Eg: ( R , + ), ( Z , + ) are groups

Abelian Group:

A group is said to be an abelian group if * is commutative in G.

ie: a * b = b * a for every a, b in G

Eg: ( R , + ), ( Z , + ) are abelian groups

Semi group:

A set together with a binary operation which is associative is called a semi group.

Eg: ( R , + ), ( Z , + ) is a semi group.

Monoid:

A set together with a binary operation which is associative and identity element exists is called a monoid.

Eg: ( R , + ), ( Z , + ) are monoids.

Finite Group and Infinite group: A group ( G, * ) is said to be a finite group if the under lying set G itself is finite otherwise G is infinite.

The number of elements in a group is called order of the group.

Addition modulo n: Let a, b be any two integers then addition modulo n of a and b is denoted by a +_n b, is defined as the smallest positive integer r, is the remainder which when obtained by dividing ( a + b ) divided by 4.

Eg: 5 +₃ 4 = 2

- 20 +₅ 6 = 0

Multiplication modulo n: Let a, b be any two integers then Multiplication modulo n of a and b is denoted by a x _n b, is defined as the smallest positive integer r, is the remainder which when obtained by dividing ( a b ) divided by 4.

Eg: 5x₄ 7 = 3

SOME EXAMPLES FOR GROUP:

Q1. Prove that G is the set of all rational numbers with the operation * defined below is a group.

Let a, b belongs to G then a * b = (ab / 2)

Proof:

(i) It is clear that if a, b belongs to G then a * b = (ab / 2) , belongs to G.

There fore * is closed in G.

(ii) Let a, b, c belongs to G then

a*( b * c ) = a*( bc / 2) = ( abc / 4 ) and

( a* b )* c = (ab / 2) * c = ( abc / 4 )
ie: a*( b * c ) = ( a* b )* c

Therefore * is associative in G.

(iii) Let x be the identity element in G then a * x = a = x * a

ie: a * x = a implies ( ax / 2 ) = a

implies x = 2 , belongs to G.

There fore x = 2 will act as the identity element in G.

(iv) Let y be the inverse of the non zero element ‘ a’ in G

Then a * y = 2 = y * a

Therefore a * y = 2 implies ( ay / 2 ) = 2

implies y = ( 4 / a ), belongs to G.

There fore y = ( 4 / a ) will act as the inverse of ‘ a ‘ in G.

Hence ( G , * ) is a group.

Example for an infinite non abelian group:

Let G be the set of all n x n non-singular matrices with real or complex entries with operation as matrix multiplication.

Verification:

(i) We have if A, B belongs to G then AB belongs to G.

Therefore matrix multiplication is a binary operation in G.

(ii) Since matrix multiplication is associative then here matrix multiplication is associative in G.

(iii) We have I, the n x n unit matrix belongs to G and AI = A = IA for every A in G.

Therefore I will act as the multiplicative identity in G.

(iv) Since every member in G is non-singular then clearly its inverse exists and belongs to G.

Therefore G is a group under matrix multiplication.

(v) Since matrix multiplication is commutative, G is abelian.

Also it is clear that number of elements in G is infinite.

Hence G is an infinite non abelian group.

Exercises:

1. Show that G = { a + bV2; a, b in Q } where V2 is the square root of 2, is a group under usual addition.
2. Show that V the set of all vectors in a space is an infinite abelian group under vector addition.
3.Show that G = {1, w, w² }where w is the cube root of unity forms a finite abelian group of order 3 under the operation multiplication.

4. Show that G= { 1,-1,i,-i } is a finite abelian group under the operation multiplication of order 4

From this section on wards in a group ( G, * ) we take a * b = ab unless otherwise stated.

Theorem ( Cancellation laws): Let G be a group and a, b belongs to G then

(i) ab = ac implies b = c ( Left cancellation law )

(ii) ba – ca implies b = c ( Right cancellation law)

Proof:

Suppose ab = ac

Since G is a group and ‘ a ‘ belongs to G, a^-1 belongs to G.

Multiply a^-1 on the left of both sides then we get, a^-1 (ab) = a^-1(ac)

implies (a^-1a)b = (a^-1a) c

implies eb = ec

implies b = c.

Similarly, we can verify that ba – ca implies b = c.

Theorem: In any group G the identity element is unique.

Proof:

Let e and e’ be two identity elements in G

Then for any ’ a ‘ belongs to G we have a e = a = e a and a e’ = a = e’a

From this it is clear that a e = a = a e’

implies e = e’

Hence identity element is unique in G.

Theorem :Let G be a group then inverse of an element ‘ a ‘ in G is unique.

Proof:

Let b and c be the inverses of ‘ a ‘ in G

Then ab = e = ba and ac = e = ca ………………………….(1)

Claim: b = c

We have from (1) ab = e = ac

Then by Left cancellation law b = c.

Therefore claim.

Hence the inverse of the element ‘ a ‘ in G is unique.

Theorem: Let G be a group and a^-1 is the inverse of ‘ a ‘ then (a^-1) ^{– 1}= a.

Proof:

Since a^-1 is the inverse of ‘ a ‘ then a (a^-1) = e = (a^-1) a

Since a belongs to G then (a^-1) belongs to G implies (a^-1) ^{– 1} belongs to G

Therefore (a^-1) (a^-1) ^{– 1} = e = (a^-1) ^{– 1}(a^-1)

We have , (a^-1) (a^-1) ^{– 1} = e implies a(a^-1) (a^-1) ^{– 1} = ea

implies (aa^-1) (a^-1) ^{– 1} = a

implies e (a^-1) ^{– 1} = a

implies (a^-1) ^{– 1}= a.

Theorem ( Invertibility): Let G be a group then (ab) ^–1 = b^-1 a^-1 for every a, b in G

Proof:

Claim: (ab) (b^-1 a^-1) = e =(b^-1 a^-1) (ab)

We have (ab) (b^-1 a^-1) = a(b b^-1 )a^-1

= a(e) a^-1

= a a^-1

= e

Similarly, we will get (b^-1 a^-1) (ab) = e

ie: (ab) (b^-1 a^-1) = e =(b^-1 a^-1) (ab).

There fore claim.

Hence (ab) ^–1 = b^-1 a^-1 for every a, b in G.

Note : The set of all remainders which when divide the sum of any two integers by ‘ n ‘ is {0,1,2,3,…, n-1}, denoted by Z _n.

Theorem: The set Z _n = {0,1,2,3,…, n-1} forms a group under the operation addition modulo n.

ORDER OF AN ELEMENT IN A GROUP

Let G be a group then order of an element ‘ a ‘ in G is the least positive integer ‘ n ‘ such that
a ⁿ = e.It is denoted by o(G).

Note: If G is an additive group then order of an element ‘ a ‘ in G is the least positive integer
‘ n ‘ such that na = e.

Eg: Consider the multiplicative group G ={1,w,w ² }, where ‘w’ is the cube root of unity.

We have 1 ² = 1 implies o(1) = 1

w ³ = 1 implies o(w) = 3

w ⁶ = 1 implies o(w ² ) = 3

Eg: Consider the multiplicative group G ={1,-1, I, -i .

We have 1 ² = 1 implies o(1) = 1

(-1) ² = 1 implies o(1) = 1

i ⁴ = 1 implies o(i) = 4

( -i ) ⁴ = 1 implies o(-i ) = 4

Subgroup:

A non empty subset H of a group G is said to be subgroup of G if H itself be a group under the same operation defined in G.

Eg: Consider the multiplicative group G ={1,-1, i, -i }.Let H = { 1, - 1 } then H is a subgroup of G.

Theorem: A non empty subset H of a group G is a subgroup of G Û

(i) (ab) belongs to H for every a, b in H

(ii) a^-1 exists and belongs to H for every ‘a’ in H.

Proof:

First suppose that H is a subgroup of G.

Then (ab) belongs to H for every a,b in H. ( By closure property in H )

Conversely, suppose that the two conditions in the statement holds.

To prove H is a subgroup of G.

By (i) closure property holds in H.

By (ii) for every ‘ a ‘ in H, a^-1 exists and belongs to H.

Since a, a^-1 belongs to H we get (aa^-1) belongs to H……………….(1)

Since a, a^-1 belongs to G we have aa^-1 = e belongs to G

Then from (1) ‘e ‘ belongs to H

There fore H is a group under the same operation in G

Hence H is a subgroup of G.

Theorem: A non empty subset H of a group G is a subgroup of G Û

(ab ^{- 1}) belongs to H for every a, b in H

Proof:

First suppose that H is a subgroup of G.

Let a, b belongs to H

Since ‘ b ‘ belongs to H and H is a subgroup, b ^{- 1} belongs to H.

Since a, b ^{- 1} belongs to H and H is a subgroup, (ab ^{- 1}) belongs to H.

Conversely, suppose that (ab ^{- 1}) belongs to H for every a, b in H.

To prove H is a subgroup of G.

We have a, a belongs to H implies (aa ^{- 1}) belongs to H

implies e belongs to H.

there fore e, a belongs to H implies (ea ^{- 1}) belongs to H

implies a ^{- 1} belongs to H.

So a, b belongs to H implies a, b ^{- 1} belongs to H

implies a(b ^{- 1}) ^{- 1} belongs to H ( By assumption )

implies (ab) belongs to H.

Hence H is a subgroup of G.

Theorem: Let H and K be subgroups of a group G then H intersection K is also a subgroup of G.

Proof:

Claim: (ab ^{- 1}) belongs to H intersection K for every a, b in H intersection K.

Let a, b belongs to H intersection K.

Then ‘ a ‘ belongs to H intersection K and ‘ b ‘ belongs to H intersectionK..

We have, a, b belongs to H and H is a subgroup implies (ab ^{- 1}) belongs to H and

a, b belongs to K and K is a subgroup implies (ab ^{- 1}) belongs to K.

implies (ab ^{- 1}) belongs to H intersection K..

implies H intersection K. is also a subgroup of G.

Theorem: For any group G the set H = { x in G / xa = ax for every ‘a’ in G} forms a subgroup of G.

Proof:

Let x and y be two elements in H then xa = ax and ya = ay for every ‘a ‘in G.

We have ya = ay implies y ^{- 1} ( ya) y ^{- 1} = y ^{- 1} (ay) y ^{– 1}

implies (y ^{- 1} y)(a y ^{- 1} ) = (y ^{- 1} a) (y y ^{– 1})

implies e (a y ^{- 1} ) = (y ^{- 1} a)e

implies (a y ^{- 1} ) = (y ^{- 1} a) for every ‘ a ‘ in G

implies y^{- 1} belongs to H.

That is y belongs to H implies y^{- 1} belongs to H.

Claim: (x y^{- 1}) belongs to H.

We have (x y^{- 1})a = x (y^{- 1}a)

= x (a y ^{- 1} )

= (x a) y ^{- 1}

= (a x ) y ^{- 1}

= a (x y^{- 1}) for every a in G.

There fore (x y^{- 1}) belongs to H.

Hence H is a subgroup of G.

Centre of a group:

For any group G the set H = { x in G / xa = ax for every ‘a’ in G} forms a group is called the centre of a group G . It is denoted by Z(G) .

That is, Z(G) = { x in G / xa = ax for every ‘a’ in G}

SYLLABUS FOR SEMESTER I & II

2K6 EN101: ENGINEERING MATHEMATICS I

MODULE - I : ORDINARY DIFFERENTIAL EQUATIONS (16hrs)

A brief review of the method of solutions of first order equations - Seperable, homogenous and linear types - exact equations - orthogonal trajectries - General linear second order equations - homogenous linear equation of the second order with constant coefficients - fundemental system of solutions - method of variation of parameters - cauchy's equation.

MODULE - II : LAPLACE TRANSFORMS (17hrs)

Gamma and beta functions - definitions and simple properties - Laplace transform - Inverse transform - Laplace transform of derivatives and integrals - shifting theorem - differentiation and integration of transforms - transforms of unit step function and impulse function - transforms of periodic functions - solutions of ordinary differntial equation using transforms.

MODULE III : VECTOR DIFFERENTIAL CALCULUS ( 18 hrs)

Functions of more than one variable – Idea of partial differentiation – Euler’s theorem for homogenous functions – chain rule of partial differentiation – application in errors and approximation – vector function of single variable – differentiation of vector functions – scalar and vector fields – gradient of a scalar field – divergence and curl of vector fields – their physical meanings – relations between the vector differntial operators.

MODULE IV : FOURIER SERIES AND HARMONIC ANALYSIS (15 HRS )

Periodic functions – Trignometric series – Euler formula – Even and odd functions – functions having arbitrary period – half range expansions – Numerical method for determining Fourier coefficients – Harmonic analysis.

DIFFERENTIAL EQUATIONS

Differential equation is an equation involving independent and dependent variables and their derivatives (or diferential coefficients). Simply we can say that differential equations means equations which involve differential coefficients.

eg: 1. ( d²y / dx² ) + x ( dy / dx ) + y = 0
2. ( d³y / dx³ ) + x ( d²y / dx² ) + y = 0
3. dy / dx = x cosx
4. ( d²y / dx² ) ² + x² ( dy / dx ) ³ = 0
5. y = ( dy / dx ) + sqrt ( 1 + [dy / dx ] )

The order of a differential equation is the order of the highest order derivative present in the equation.
In the above examples order of equations (1) & (4) are 2 where as equation (3) has 1 & (2) has 3 .

The degree of a differential equation is the degree of the highest order derivative present in the equation, after the equation is free from radicals and fractions.
In the above examples degree of equations (1) ,(2) & (3) is 1 where as equation (4) & (5) has 2 .

A differential equation is said to be linear if it involves dependent variable and its derivatives, occcur in first degree.

The solution of a differential equation is the relation between the independent and dependent variables, not involving its derivatives such that it should satisfy the differential equation.

SOLUTIONS OF FIRST ORDER FIRST DEGREE DIFFERENTIAL EQUATIONS

The general form of a first order first degree differential equation is M dx + N dy =0 where M and N are functions of x and y. Now we procced to develope the solutions of the equations in some particular cases when M and N are certain functions of x and y. So we put forward some methods to solve such differential equations.

1. VARIABLE SEPERABLE FORM
A first order first degree differential equation is said to be in variable seperable form if it is of the form M dx + N dy =0 where M and N are respectively functions of x and y alone.

Problems:
1. Solve ( dy / dx ) = (x² / 1+ y²)