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4 Maths -- Elementary Group Theory

Determine the identity element and inverse elements of 3 and -2,Given an algebraic structure (Z, ౦) with binary operation ౦ defines by m ౦ n = m + n  + 1 for all m, n ∈ Z.

Determine the identity element and inverse elements of 3 and -2,

Given an algebraic structure (Z, ౦) with binary operation ౦ defines by m ౦ n = m + n  + 1 for all m, n ∈ Z.

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Let e be the identity element of 3 under binary operation defined by, 

m ౦ n = m + 1 + 1.

Then, 3 ౦ e = 3

or, 3 + e +  1 = 3    [m * n = m + n + 1]

So, e = -1.


Again, let e’ be the identity element of – 2.

Then, (-2) ౦ e’ = - 2.

or, (-2) + e’ + 1 = -2

or, e’ + 1 = 0

So, e’ = - 1.

So, - 1 is the required element of both 3 and – 2.


Again, let i be the inverse element of 3 under the given binary operation ౦.

Then, 3 ౦ i = e

or, 3 + i + 1 = - 1    [m * n = m + n + 1, e = -1]

or, i = - 1 – 4 = - 5.

So, - 5 is the inverse element of 3.


Finally, let i’ be the inverse element of – 2.

Then, (-2) ౦ i’ = e’

or, (-2)  + i’ + 1 = - 1.

or, - 1 + i’ = - 1.

So, i’  = 0

So, 0 Is the inverse element of – 2.


More questions on Elementary Group Theory

S is a given set and a,b ∈ S. Prove that the operation ∗ defined by a ∗ b = 3a +5b on set S= set of positive integers, is a binary operation.

Here, S is a set of positive integers.

a b = 3a + 5b

The multiplication and addition of integers is always a positive integer.

Thus, for all a, b S, a b = 3a + 5b S

So, it is a binary operation.



Test the closure, associative and communitative properties for the operation defined by m ∗ n on Z, m,n ∈ Z.

- For closure, let m,n Z,

m n = n Z.

So, it is closed.

- For associative, let m,n and p Z.

(m * n ) * p = n * p = p Z

m * ( n * p) = m * p = p Z

(m * n ) * p = m * ( n * p)

So, the operation is associative.

-For communitative, let m,n Z.

m * n = n Z

n * m = m Z

m * n n * m

So, the operation is not communitaitve.





Given a set  Z= { 0, 1, 2, 3 } and a binary operation +4 is defined by the following Cayley’s table.+4012300123112302230133012Find the identity elements and the inverse elements of 2 and 3.

Here, Z= { 0, 1, 2, 3 }


+4

0

1

2

3

0

0

1

2

3

1

1

2

3

0

2

2

3

0

1

3

3

0

1

2


Identity element of 2 is 0 because 2+40 = 2 and 0+42 =2

Identity element of 3 is 0 because 3+40 = 3 and 0+43 =3

Inverse element of 2 is 2 because 2+42 = 0

Inverse element of 3 is 1 because 3+41 = 0

A binary operation * defined on the set S= {a,b,c} is presented in the following Cayley’s table.*abcaabcbbcaccabShow that (S,*) forms a group.

From the table,

*

a

b

c

a

a

b

c

b

b

c

a

c

c

a

b

We see that the operation defined on any two elements of S gives an element of S itself.

i.e. For each a,b S, a*b = S

So, S is closed under operation *.


For each a,b S,

a * (b * c) = (a * b) * c

or, a * a = b * c

or, a = a (true)

So, * satisfies associative property.


a * a = a,

a * b = b * a= b

and, a * c = c * a = c

a is an identity element.


a * a = a,

a is the inverse of a.

Similarly, b * c = c * b = a,

b and c are the inverse elements of c...

A set of matrices of the formwhere θ is a number, is given. Show that the operation of matrix multiplication is closed.

Here,




which is closed.

If the group (G, ౦) is communitative show that, (a ౦ b)-1  = a-1 ౦ b-1, for all a, b ∈ G.

Let a-1 and b-1 be the inverse elements of a and b in the communitative group (G, ) with the identity element e, then,

a a-1 = e = a-1 a

b b-1 = e = b-1 b


(a b) (a-1 b-1) = (a b) (b-1 a-1)

= a (b b-1) a-1

= a e a-1

= e (a a-1)

= e e

= e

Similarly, (a-1 b-1) (a b ) = e

Hence, (a b)-1 = a-1 b-1

S is a given set and a,b ϵ S. Prove that the operation * defined by a*b= 3a+5b on S=set of positive integers, is a binary operation.


Test the closure, associative and commutative properties for the operation defined by m*n=m+n+1 ,m,n ϵ Z.


Let G= {0,1,2} form a composition table for G under addition modulo 3. Find the identity element and inverse element of 1 and 2.


Show that the set S= {-1,0,1} does not form a group with respect to addition operation


Show that the set of all positive rational numbers forms an abelian group under the composition a*b=ab/4.


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