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.