Identity is only Idempotent Element in Group

Theorem
Every group has exactly one idempotent element: the identity.

Proof

 * The identity is idempotent.
 * From the Cancellation Laws, all group elements are cancellable.
 * If $e$ is the identity of a monoid $\left({S, \circ}\right)$, then $e$ is the only cancellable element of $S$ that is idempotent.