Identity of Group is Unique

Theorem
Let $\left({G, \circ}\right)$ be a group which has an identity element $e \in S$.

Then $e$ is unique.

Proof
By the definition of a group, $\left({G, \circ}\right)$ is also a semigroup with an identity element.

The result follows by applying the result Identity of Semigroup is Unique.