Question

Prove that if G is a group then identity element of G is unique​

Answer 1

Answer:

to prove that the identity element of a group is unique, assume G is a group with two identity elements e and f. Then e = e ⋅ f = f, hence e and f are equal.

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Answer 2

Answer:

If G is a group then identity element of G is unique​

Step-by-step explanation:

Let G be a group.

Let both e and e' be the identities of G

Then, ae = a for all values of a in G                 --(i)

and e'a= a for all values of a in G                    --(ii)

If we put a = e' in equation (i),

we get, e'e = e'                                           --(iii)

On substituting a = e in equation (ii),

we get, e'e = e                                           --(iv)

On equating equation (iii) and (iv), we get, e' = e

Therefore, they both are equal

=> e and e' are same

=> The identity element of a group is unique.

Hence proved.