Question

Define a group prove that the identity element of a group G is unique in G

Answer 1

Answer:

As noted by MPW, the identity element eϵG is defined such that ae=a∀aϵG. While the inverse does exist in the group and multiplication by the inverse element gives us the identity element, it seems that there is more to explain in your statement, which assumes that the identity element is unique.

Answer 2

Answer :

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 .

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 .