Question

Prove that the identity element of a group is unique.

Answer 1

As noted by MPW, the identity element $eϵG$ is defined such that $ae=a\forall aϵG$

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.

A more standard way to show this is suppose that $e$, $f$ are both the identity elements of a group $G$.

Then, $e=eof$ since $f$ is the identity element.

= $f$ since $e$ is the identity element.

This shows that the identity element is indeed unique.