Identity is only Idempotent Element in Group

Theorem

Every group has exactly one idempotent element: the identity.

Proof 1

The Identity Element is Idempotent.

From the Cancellation Laws, all group elements are cancellable.

The result follows from Identity is only Idempotent Cancellable Element.

$\blacksquare$

Proof 2

Let $\struct {G, \circ}$ be a group whose identity is $e$.

Let $x \in G$ such that $x \circ x = x$.

\(\ds e\)

\(=\)

\(\ds x \circ x^{-1}\)

Group Axiom $\text G 3$: Existence of Inverse Element

\(\ds \)

\(=\)

\(\ds \paren {x \circ x} \circ x^{-1}\)

by hypothesis: $x \circ x = x$

\(\ds \)

\(=\)

\(\ds x \circ \paren {x \circ x^{-1} }\)

Group Axiom $\text G 1$: Associativity

\(\ds \)

\(=\)

\(\ds x \circ e\)

Group Axiom $\text G 3$: Existence of Inverse Element

\(\ds \)

\(=\)

\(\ds x\)

Group Axiom $\text G 2$: Existence of Identity Element

$\blacksquare$