Let G be a group prove that G has a unique identity element

Let G be a group prove that G has a unique identity element. Hint: Suppose e and e\' are both identity elements then consider e*e\'

Solution

Let G be a group. Suppose that G has two identity elements e and e\'.

Then, by definition of identiy, when e is an identity

e\'.e = e (1)

when e\' is an identiy

e.e\' = e\' = e\'.e (2) { since a.e=e.a=a }

from (1) and (2), we get

e = e.e\' = e\'

Hence, e = e\'

Therefore, identity element if group G is unique.