Question

Prove that the identity element e of a group G is unique.

Answer 1

An identity element can be a left-identity (1*x=x, x∈R) a right-identity (a^1=a, x∈R) or both (1*x=x, x*1=x, x∈R).

If the group (S,*) is such that it has a left (e1) and right (e2) identities, then it follows that
e1*x=x ∀x∈S, and
x=x*e2 ∀x∈S

From which we conclude:
e1=e1*e2=e2 meaning that e1=e2, or the (two-sided) identity element must be unique.