Identity 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.

$\blacksquare$

Sources

• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 1.4$: Lemma $1$