Question

Prove that in a group G, there is only one identity element.

Answer 1

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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 :

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 a group G .

Since e is the identity element , thus we have

e * e' = e' * e = e' ........(1)

Also ,

Since e' is the identity element , thus we have

e * e' = e' * 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 .