Question

The identity element and the inverse of an element in a group (g , o) are unique

Answer 1

Explanation:

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Answer 2

Note :

• Group : An algebraic system (G,*) is said to be a group if the following condition are satisfied :

1. G is closed under *
2. G is associative under *
3. G has a unique identity element
4. Every element of G has a unique inverse in G

• Moreover , if a group (G,*) also holds commutative property , then it is called commutative group or abelian group .

Solution :

1) To prove :

In a group G , there is only one identity element .

Proof :

Let's assume that , e and e' are two identities elements of the group G .

Now ,

Since e is the identity element , thus we have

e o e' = e' o e = e' ........(1)

Also ,

Since e' is the identity element , thus we have

e o e' = e' o e = e .........(2)

From eq-(1) and (2) , we have

e = e'

Hence ,

The identity element of a group G is unique .

ie. The group G has only one identity element.

Hence proved .

2) To prove :

The inverse element of a group is unique .

Proof :

Let a be any arbitrary element of a group G and let e be the identity element in group G .

Let's assume that , b and c are two inverses of a .

Then , we have

a o b = b o a = e and a o c = c o a = e

Now ,

b o (a o c) = b o e = b

(b o a) o c = e o c = c

But in a group , the associative property holds .

Hence , we have

b o (a o c) = (b o a) o c

→ b = c

Hence ,

The inverse of an element in a group is unique .

Hence proved .