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}